So, I'm not sure how much background you have, but I think it's first important to understand the technical aspects of what gauge is and how it works and why it's important. Gauge is the distance between the inner faces of the rails. It sets the spacing of the wheels, length of the axles, etc. This means that a single railcar, whether a locomotive or an unpowered carriage, can only run the gauge it was built for. There are some clever workarounds today, such as variable-gauge axles (where the train is driven through an apparatus changing the length of the axles) or swappable bogies (where the whole wheel/axle assembly is swapped out). The latter is time consuming, and both are rather complicated and were developed after the era you're asking about anyway. It's fairly straightforward to redesign a locomotive or car for a new gauge, so there can be commonality in design, but it's often impractical to actually move the same train from tracks of one gauge to a different one in regular service.

One last piece of background--standard gauge is 4' 8.5". It's the common gauge in the US, Canada, much of Europe (not Spain, Ireland, or the former USSR), China, and elsewhere too. Gauges wider than this are referred to as "broad gauge", which includes Russian gauge (1520mm), Indian gauge (5' 6"), Irish gauge (5' 3"), and Iberian gauge (1668mm). Gauges narrower than this are called "narrow gauge", the most common of which are Cape gauge (3' 6") and meter gauge (1 meter, as the name suggests). But, that's the current situation--standardization of gauges didn't happen all at once. For a while railroads had their own gauge in the US. During the Civil War most of the Southern Railway network was broad gauge, while the North was mostly standard gauge. Even today there are occasional holdouts in urban rail systems--a few lines use Philadelphia Trolley Gauge, and streetcars in Toronto have their own gauge, for example.

So in the mid-19th century in the US, there were a number of competing gauges in common railway use. Places where these met are called breaks-of-gauge. Erie was one of these places. The mechanics of gauges meant that passengers wishing to travel from one rail line to another via Erie would have to disembark their train and catch a different one. While a huge inconvenience for passengers (not only would they have to switch, but there'd be a layover, and if their train into Erie was late, they might miss their train out), this was beneficial for Erie residents, since they had a reliable source of tourists, essentially, who had no choice in stopping in Erie. Standardizing the gauge took this benefit from Erie, and the local government resisted this, as you can see.

So for railroads, standardizing gauges was a big deal because it allowed the same trains to operate over different railroad's systems. Not standardizing gauges was a big deal for Erie, because not having a standard gauge meant that they had a source of reliable revenue in the form of tourists.

It's important to consider just how important railroads were in this period- imagine a combination of the interstate highway system, airplanes, and the Internet (for telegraph lines tended to follow the railroads) rolled into one. Now consider that this system took vast amounts of labor, time, and money to complete, but it was flawed in that it was built by a hodgepodge of concerns, and caused massive inefficiencies by forcing a change of conveyance at many points, for it was really more a great number of systems that interconnected, rather than one network. The clear solution would be to standardize the whole shebang, but this would require yet more labor, time, and money, the costs of which would be borne by whoever's standard didn't win. History tells us that format wars can be vicious and lengthy (consider AC vs. DC, VHS vs. Betamax, HD-DVD vs. Blu-Ray, the vast panoply of competing standards for device charging, etc., etc.), and this conflict underpinned the entirety of the U.S. economy. Basically, there was a huge amount at stake, it was in every stakeholder's best interest to avoid compromise, and everyone else was getting shafted by the failure to reach a solution.

Variety of reasons, sometimes quite arbitrary. As with any new technology everybody thought their standard was the best and would conquer the world. And at times, standartization was opposed because of fears that a neighbour could invade via the railways. The difference in gauge prevented the enemy transport appearing unexpected.

When George Stephenson developed a successful locomotive, the company where he worked build lines accordingly. The locomotive got used by a large railway company, S&D, which kept the same gauge. S&D build the first intercity line in England, and with that being a success, the gauge became "standard" choice throughout the country.

Until another railway company employed a hugely innovative engineer named Brunel. Brunel had tons of inventions and being widely influential, introduced a new gauge, to give the train much greater stability. Thus we have the "broad" gage.

Then the "standard" gauge got exported to other European and North American countries. But the different railway giants were competing for coverage and each was building a different gauge lines. When a new line was proposed, the gauge size would determine the allegiance to one company or the other. It was a matter of conquest and being different in order to push the competition out of business, rather than a matter of cooperation. In England, it took an Act of parliament to begin gauge conversion to "standard" size.

In Australia, the chief engineer to the Sydney Railaway Company was Irish and thus he persuaded the company to adopt the Irish size of gauge. When he resigned after dispute over pay, the company replaced him with a Scottish engineer, who had a preference for the English size gauge. And thus, he used his influence to introduce a change and request alterations. At the same time, other parts of the country adopted different gauges since their networks were not connected. The country was so big and fragmented that the variety in gauge size was not an obvious practical issue until later.

In another part of the world, Portugal and Spain had different gauges due to the difference in measurements. Spain used Castillian feet and Portugal measured its rail in Portuguese feet. This eventually lead to the creation of a new gauge as a compromise between the two, which finally allowed interoperability.