First, I'll speak to how this related to Major League Baseball, the oldest of the North American professional sports leagues and, at least in this case, the most important for our story, as MLB was the pioneer that the other leagues followed (more or less).

The long and short of it is because of free agency. Free Agency allows players to sell their talent to the highest bidder, the process of which drives up the prices of the average player, such that today, the average MLB salary is above $3 $4 million per season. But it wasn't always this way.

It used to be that there was a "reserve clause", which stated that after a contract expired, the rights to the player remained with that player's team. This was pro forma for all teams and the entire systems is usually referred to as the "reserve system". This clause had been in force since the very beginning, at least since the 1890s (the modern era of the game is said to have begun in 1901), and it is what kept players playing for the same team because a player's only option was to quit playing professional baseball altogether if they did not want to accept the salary that the team dictated to them.

Now as you can imagine, the very best players were not as heavily affected by this because teams were hesitant to lose out on the revenue those players generated, since they were, even with inflated salaries by contemporary standards, big money makers, and teams did not want to take the risk of having such star players simply not play at all. This exception is best encapsulated by the Sultan of Swat, Babe Ruth, arguably the best player to have ever played the professional game. In 1930, Ruth signed a contract paying him $80,000 a year, which was actually more than the President earned that year. Adjusted for 2012 dollars, it works out to about $1.4 million. For the entire reserve era, no other player would come close to matching that amount in adjusted dollars, although the other person generally argued to have been the best player ever, the Say Hey Kid Willie Mays, earned $160,000 in 1959, which works out to more than $1.2 million in adjusted 2012 dollars. But these superstars were really the exception to the rule, as most players were not good enough to command the contracts of a Mays or a Ruth. Instead, they languished in the reserve system, forced to except the paltry salaries offered in their contracts or not play at all. This is why you saw players who had off-season jobs designed to supplement their income.

All of this began to change in 1969. That year, outfielder Curt Flood was traded from the St. Louis Cardinals to the Philadelphia Phillies. Flood protested to the Commissioner, Bowie Kuhn, arguing that he was entitled to hear offers from other teams before accepting the trade to Philly. Kuhn denied his request, so in response, Flood filed a federal lawsuit, which eventually reached the Supreme Court in 1972. The court actually sided against Flood, but his lawsuit had sort of let the genie out of the bottle, so to speak. In the beginning of the 1972 season, the players went on strike for the first time in MLB history. The strike lasted only a couple of weeks, but as a result of the bargaining to get the players back onto the field, the owners acquiesced to the demands of the MLB Players Association and instituted a system of salary arbitration, in which disagreements between the player and the team would be handled, not by the court system, but by private arbiters. After the 1974 season, two players, Andy Messersmith and Dave McNally refused to sign the reserve contract offered by their respective teams, playing out the 1975 season without a signed contract. After the 1975 season ended, both players filed grievances with MLB arguing they should be free agents. Because of the system of of salary arbitration, the disagreement went in front of a group of three arbiters, one selected by the players, one by the owners, and one mutually agreed upon, Peter Seitz. On Dec. 23, 1975, Seitz, who was necessarily the deciding vote on the three man panel, ruled in favor of the players, effectively overturning the USSC decision in Flood v. Kuhn and ending the reserve system in one fell swoop. McNally and Messersmith were granted free agency status and allowed to sign with any team at a price mutually agreed upon. McNally actually retired from the game, but Messersmith signed with a new team for the 1976 season, the Atlanta Braves, marking the first time an already established player was allowed to sign with a team of his own choosing for a mutually agreed upon salary. The following year, MLB and the Player's Association negotiated a new collective bargaining agreement which allowed for all players to reach free agency after six years service in the major leagues, a system that is still in place today.

Because of free agency, player salaries have gone up considerably, but I would caution against any notion that they are overpaid (they're not). Despite the large increases in salary, especially of the best players, the average salary in baseball has not kept up with increases in the revenue the teams generate. This is not to say that the players are not better off than before free agency - they certainly are - only to say that the players have not reaped the benefits of the larger pie they have created as the decades have passed.

A couple of sources that speak to all of this: 1 2 3

EDIT: just a little prose to make things read more clearly.

EDIT 2: I should mention that MLB tried to file a lawsuit in federal court to overturn the Seitz decision, but failed. Free agency along the MLB model followed in other major North American sports leagues in the 1990s (NFL, 1992; NHL, 1995; and NBA, 1996).

EDIT 3: I've gotten a lot of questions about whether the increased total revenues of MLB are really the reason for increased player salaries. The idea here is that because the total pie has gotten larger, the player's salaries have grown. The evidence, however, shows that although the pie has certainly gotten larger, the real driver of increased salaries is not the pie getting larger, but player's getting a larger piece of the pie. Below is a chart of the MLB's average payroll as a percentage of team revenues. There is a clear uptick in the percentage of team revenues awarded to the players via salaries after Free Agency became standard in the mid 1970s. Source

Year-Percentage
1929 - 35.3
1933 - 35.9
1939 - 32.4
1943 - 24.8
1946 - 22.1
1950 - 17.6
1974 - 20.5
1977 - 25.1
1980 - 39.1
1985 - 39.7
1988 - 34.2
1989 - 31.6
1990 - 33.4
1991 - 42.9
1992 - 50.7
1994 - 60.5
1997 - 53.6
2001 - 54.1

Professional baseball player here. Just wanted to add my two cents

First off let me start by distinguishing the differences in the levels of pay in professional baseball

All of these levels are only paid seasonally (5 month season)

-Minor leagues-

Rookie ball $1,100 -a month Low A $1,150 High A $1,200 Double A $1,500 Triple A $2,300

-Major leagues-

$500,000+ a year minimum

Now most minor leaguers DO have jobs every off season since as you can see the pay is very poor. The select few who don't need off season jobs are the guys who signed big bonuses which is only about 150ish guys out of the 1200ish of every years draft.

Now to make it to the major leagues is an incredible accomplishment. At any given time there are only around 700 major leaguers in the world. When one gets promoted another gets demoted.

The ones who have stayed healthy, played well enough, and maybe had a little luck, will sign a contract for guaranteed money. The other guys who are just playing in the bigs making league minimum only get paid big money for the days they are actually on the big league roster.

Teams often send players up and down all year and use players like pawns in a chess game. They use a certain player for a day here or there when needed otherwise they're back in triple A.

More often then not when players are injured teams will release them once healthy because that player has become a liability to the team. No team wants to pay someone who can't contribute. If a guy gets injured in the big leagues he is guaranteed big league minimum until healthy. So it pays to be a big name guy because if you're an average joe the team has no need to keep you. With the abundance of free agents from around the world.

So if players can afford it they would prefer to spend their off seasons training as opposed to working real jobs in the off season.

Two developments in the seventies turned NASCAR racing into a big-money sport, with drivers' salaries to match: the ban on tobacco advertising on television, and the founding of ESPN.

When cigarette commercials were outlawed in the early seventies, tobacco companies turned to sponsoring sporting events instead. R.J. Reynolds, based in North Carolina - stock car racing country - took over sponsorship of NASCAR's highest tier of racing, which became known as the Winston Cup Series. The company also put up a lot of prize money, including the "Winston Million" - a million dollar prize for the driver who wins the highest profile races in a single season. (Bill Elliott won it in 1985 - though he ultimately lost the championship that season.)

NASCAR racing got some network TV coverage in the seventies, but races were rarely if ever shown live. When ESPN was founded in 1979, the fledgling company needed product, and most of the major professional sports leagues were simply out of the company's reach. NASCAR racing turned out to be a good fit, and the network showed most races live through the seventies and eighties - building a dedicated fan base outside of the South. That in turn led to more big-name sponsors and merchandising, and by the turn of the millennium NASCAR had become a big, extremely lucrative business.

Nobody's done a detailed writeup of how it happened in football/soccer yet, so if I may:

The modern history of English football begins in 1888 with the formation of the Football League (not to be confused with the slightly older Football Association). As you might expect for a game whose basic rules have only recently been codified, early FA-sanctioned matches are a bit chaotic, with teams setting their schedules pretty much independently. The only real extent to which the FA had any control in this regard was in its hallmark competition, the FA Cup - initiated in 1872 and still contested to this day. The League is formed in 1888 to deal with some of these problems - namely to ensure a more uniform means for competition between clubs. They also enshrine professionalism in the sport, though the FA quietly did so 3 years earlier.

Now, when the FA legalizes professionalism it also makes another ruling: all professional players must be registered to their club and cannot play football in any other capacity except the one to which they're registered. Prior to this you had cases of semi-pro players playing paid for one side and free for another, usually smaller, team. Registration has to be renewed annually - no registration, no play. In the 3 year period between professionalism and the League's formation, you do see something like a "free agency" crop up - when a player's yearly registration expires, he can go wherever he'd like. This of course benefits the bigger, richer clubs such as Sunderland and Aston Villa (ironically, two of the worse off clubs these days) who can basically buy up whomever they want. The League sees this as a bad thing, and institutes essentially a contract-within-a-contract system - the "retain-and-transfer" rule. This states that, rather than registering annually, registration is indefinite unless terminated by the club - what it notably doesn't explicitly say is that there's no requirement that registered players be given contracts. So you have a scenario now where players have two options: take what's offered, or sit in limbo unless you can convince your team to cancel your registration. Not unlike the MLB prior to the 1960s, as discussed earlier. What ends up developing is the concept of the "transfer fee" - essentially a royalty paid by one club to another in exchange for the holding club terminating a player's registration. This remains a major feature of European football - when you see figures bandied about in the millions of pounds/euro range, these are transfer fees rather than payment to the player himself.

What really kickstarts extreme transfer-fees and high-profile pay (though some players, like Pele and David Beckham, are paid very well prior to this) is the Bosman Ruling. In 1990, Belgian star Jean-Marc Bosman's contract is ending and he wants to move to a team in France once it concludes; the French team doesn't want to pay the 500,000 pound transfer fee his current employers put on him. Under the retain-and-transfer system, he's stuck; what's more, his wages get slashed due to not playing. Bosman challenges this in the courts and after a 5 year legal battle the European Court of Justice rules that the end of a contract also terminates any registration - the "retain" portion of retain-and-transfer is dead.

After this you start to see transfer fees and wage payments balloon in scope - no club wants to be caught in a Bosman-like situation where its assets can walk without them seeing a cent, so expensive, lengthy contracts get offered to players to keep them on-board long enough for the club to turn a profit on via a transfer. The exact size of contracts in European football is significantly less transparent than player payment in American sports, but by way of example one of the highest-paid players in Europe, Zlatan Ibrahimovic, is rumored to pull in 300,000 pounds per week - compare that to the 500,000 offered to Jean-Marc Bosman's club (as opposed to the man himself, who likely made far less) and you can definitely see how quickly things have come along.

In Soccer the first professional teams were formed in England around 1885. The first continental professional championship was the Austrian Championship in 1924/25, after the Anschluss (Joining) of Austria to Germany professional Football was forbidden by the Nazi Government, even while german teams were paying their players an illegal salary or even tried to convince players to switch to their teams. In Germany the DFB allowed so called Vertragsspieler in 1949, in 1963, with the start of the Bundesliga they began giving out player licenses, which every player today needs to play professional football, but still players were calle semi-proffesionals, as the DFB only allowed a maximum Salary of around 1.200DM. It was only in 1972 the DFB allowed players to become full professionals, without a maximum salary.