Gosh, what a great question! It's such a great question, in fact, that there is no definitive answer to it. (Maybe you can be the person who trawls the Nixon tapes and finds the answer.)
Before I start the discussion, it might be worth pointing out something that you (well, maybe not you in particular /u/TrynnaFindaBalance, but you collectively, as in everyone reading this) might have overlooked.
What's obvious to us today, with the benefit of hindsight, was not clear to Nixon or anyone else in the United States in early 1972. We know the results, and so we have an artificial assurance that because things turned out they way they did, they must have turned out that way. Living in the moment is to live in uncertainty.
At the start of 1972, opinion polls showed Nixon and the various Democratic contenders in a close contest. Sen. Edmund Muskie was considered a strong candidate, and Nixon feared facing Alabama Gov. George Wallace (who was ultimately incapacitated in a May 1972 assassination attempt).
At the start of the year, when G. Gordon Liddy and Attorney General John Mitchell discussed the campaign to harass and sabotage the Democratic Party, Nixon's wide victory wasn't yet apparent. That would come later.
It's also important to remember that the June break-in — for which Gonzales, Barker, McCord, Martinez, and Sturgis became infamous — was only part of the campaign approved by Nixon's backers (including Liddy). That campaign also included wiretapping and other kinds of snooping. Remember that Nixon himself may not have been aware of the exact activities: It was the coverup that got him in trouble.
But that doesn't get to your key question: "What were the burglars after?"
There are a ton of fun theories about this. The most popular theory is that Nixon believed that Democratic National Committee chairman Lawrence O'Brien had some dirt on a loan given by billionaire Howard Hughes (yes, of Spruce Goose and "The Aviator" fame) to Nixon in 1969. Some of that money, according to reporting by "60 Minutes," went to paying for frills at Nixon's Key Biscayne home.
E. Howard Hunt, who hired the burglars, told the Miami Herald in 1997 that they were actually looking for ties between the Democratic National Committee and foreign governments. They were searching for evidence of foreign campaign contributions (something that might be familiar to Americans lately), which would make great dirt in the election that year.
“What we were looking for is the same thing every congressional committee is looking for today, which was evidence of illegal foreign contributions,” he told Herald. “That was the rationale for going in there. We’d heard rumors that both the Vietnamese and Fidel Castro were inserting funds illegally into the Democratic National Committee. And the idea was to look at the books, photograph them, in and out, and that’s it. It didn’t seem like such a deal to me. You know, I’d been doing that stuff for years, a ‘black-bag job’ into other embassies. But you know, I didn’t have skilled people.”
In 2001, Liddy testified that the burglars weren't after anything political: They were actually looking for photographs that put the wife of White House counsel John Dean into a call-girl ring.
Contemporaneous rumors held that the motive behind the burglary was to find evidence held by the Democrats that linked Nixon to various assassination attempts made by the United States against Fidel Castro. A lot of those rumors stemmed from the fact that the Watergate burglars had deep ties to anti-Castro plotters, and some Americans reveled in the idea that the nation's dirty tricks had come home to roost.
Columnist George Will has an interesting theory that ties the motive to events happening in the days surrounding the burglary. One year before the break-in, the New York Times began publishing the Pentagon Papers. Will has suggested that Nixon was worried that the Papers would expose Nixon's 1968 role in sabotaging peace talks related to the Vietnam War. Nixon's successful sabotage of those talks, revealed only in the past few years, violated the Logan Act of 1799.
Regardless of the motive, it seems likely that Nixon himself had no particular knowledge of the June break-in, but he did know broadly about a campaign of "dirty tricks" against the Democratic Party to hamper Democrats' efforts to fight him in 1972. And as I said before, it wasn't the break-in that ultimately forced Nixon to resign: It was the cover-up.
That's something worth remembering.
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It probably helps to split the question up: Why would the Nixon team do something like that and why would they do exactly that?
The first question isn't that difficult to answer when you look at the operations that immediately before that. There was the break-in into the office of Daniel Ellsbergs psychiatrist, the attempt to dig up (more) dirt on the Chappaquidick incident, the forged Diem cables to further damage the Kennedy government and more dirty tricks.
That was driven by two factors: a) the belief that the democrats were playing as dirty as they were and that such tricks had cost Nixon the presidency in 1960 and b) the general conviction that looks mattered more issues when it came to politics. There are many examples for both points on the Watergate tapes. Nixon was convinced that the Democrats had bugged his campaign in 1960 and employed Ex-FBI agents to sweep his hotel rooms and offices when he ran in 1968. And there are many quotes by Bob Haldemann, Nixons Chief of Staff and Ex-advertisment executive that show how focused the White House was with the public opinion:
We failed to move on to the PR stage. We should have been able to make the PR case, at least, even though we couldn’t carry the legal case. [...] We need a strong offense. The question is – can we sell it [...]. Our whole concern now must be the public aspects of this. The morality issue is moot because the papers are already out.
(Haldemann Action Paper: The New York Times Papers. June 15, 1971, in: Bruce Oudes: From the President. Richard Nixon's secret files. New York 1990, 271.)
The explanation that the Watergate break-in was in character for the Nixon administration is probably not very sexy argument, but it is the approach to understand it. Nixon and his closest advisers fostered a siege mentality in the White House and sincerely believed that it was them vs the corrupt Eastern establishment. This deep mistrust to the established institutions led directly to the absurd secrecy and clandestine projects that brought his administration down.
Now, why did they target the Watergate and the DNC? That's way more difficult to answer. There are tons of stupid conspiracy theories about this. One says that the wife of Nixon advisor John Dean had been working as an escort. Larry O'Brian, the DNC chairman had evidence of that in his files, so John Dean ordered the break-in to get these photos. There's no good evidence for that and all those other theories that say the Nixon team was searching for some specific informations in O'Brian's files: Evidence of bribes to Nixon from Howard Hughes, explosive information concerning the Kennedy assassination, ...
Here, again, the best explanation is rather boring: Howard Hunt and Gordon Liddy went (kinda) rogue and organized the break-in on their own. It's like the assassination of Thomas Becket: Will no one rid me of this turbulent priest?
Hunt and Liddy were hired because the White House was very dissatisfied by the recent work of the FBI. Hoover was old and ailing, and the FBI saw itself as a weapon against communist agents and traitors; most of its agents didn't care or understood the political infighting in Washington.
Nixon repeatedly railed against disloyal government employees and officials that had been appointed by the JFK and LBJ administrations. The antiwar movement was undermining the government, the media was completely controlled by these forces, and the FBI was just worried about foreign agents directly controlled by Moscow. Gordon Liddy wrote several memos that directly attack the wrong focus of Hoover and the FBI:
Today it is the Ellsbergs in sensitive positions in Government we must contend with. The past 15-20 years have witnessed the evolution of a new breed of fanatics.
(Memorandum Re: Organizational Survey, July 23, 1971, in: WHSF-SMOF: David Young, Box 13, Folder: Office Chron File of G. Gordon Libby, RNPL)
The Nixon administration had tried to integrate the FBI in the White House (the infamous Huston Plan) but Hoover had prevented that and kept the Bureau independent. So Nixon's advisers created their own team for domestic intelligence operations, the Plumbers. One of their founding members, Egil Krogh, wrote later:
Liddy’s report to us about [the] deficiency [of the FBI] was an important factor in the self-help mentality that then began to emerge in the unit. Rather than questioning the Bureau’s lack of support, we resolved to go forward on our own with this investigation.
(Egil Krogh: Integrity. Good People, Bad Choices, and Life Lessons from the White House. New York 2007.)
This self-help mentality is important to understand the actions of the Plumbers.
I think it's save to say that Nixon didn't know about the break-in and hadn't ordered it. Hunt and Liddy hadn't thought the operation through, there was no direct goal and specific plan to follow. It was born out of the general belief that something had to be done. That the upcoming election was looking bright wasn't the point: Nixon and his advisers worked under the perception that the general public, the political establishment and the officials in Washington were against them. You can't govern efficiently under these conditions, which the next election won't change. So you gotta help yourself.
Nixon didn't knew about the Ellsberg break-in beforehand, but approved it afterwards. The general goal was to find evidence that would damage Ellsbergs reputation in the public. But the Plumbers found nothing, because psychiatrist usually don't keep their notes in these office files. The whole burglary was poorly planed, so-so executed, and yieled absolutly nothing of value at all. It was done by the same personal that a couple of weeks later got caught inside the Watergate, so it's probably save to draw a few conclusions from that.
Several people (Haldemann, Ehrlichman, Coulson) wrote in their memoirs that they pretty quickly recognized Watergate as the foolish third rate burglary that it was. I think that's written with the benefit of hindsight. Would they've come to the same conclusion if Liddy and Hunt didn't get caught? I don't think so. Watergate was too much in character for the Nixon administration. Limited upside, a lack of clear goal and not fully thought through plans hadn't stopped their other operations either.