I have seen a lot of beers that have been corked and I can't really understand what the significance is. When I tried to look it up, most of what I found just related to how to properly store corked beer and similar means, but nothing really answering my question. I am trying to figure out what corking a beer actually does to it and if there are any real benefits. Is it simply a style choice or is it there for a reason, like it is for wine bottles. Does it actually have an effect on the beer's taste, quality, etc?
The likely answer is somewhere between carbonation pressure and marketing.
It's hard to find numbers for how much caps can handle vs corks but you should notice that most corked beers also come in bottles with very thick glass, this is because the beer inside is at a higher pressure than most other styles.
Most beer styles will fall pretty close to 2.0 or 2.5 volumes of CO2 (just the term, think of the number as a relative baseline), but a lot of Belgian strong beers like Bierre de Garde or Tripel will be carbonated with 3.3+ volumes. Lambics and other sour beers can go up to 4.5 volumes. This is a linear scale so that becomes possibly twice as much pressure as a normal bottle...so yeah, thicker glass.
At such high pressure you might find corks will stay put more reliably than a cap since it has more surface area in contact with the bottle. It also might provide a more airtight seal than a cap for long-term aging.
Those would be technical considerations.
Corking also predated capping so there's also a very traditional feel about a bottle with a cork and a cage, which can fit into the traditional image a lot of breweries market themselves using. I say I'm not sure and it could just be marketing because a lot of Wheat styles like German Wheats, Belgian Wits, etc are fairly highly carbonated (near the 3.3 volumes range) but most of them have no problems being capped.
As to the affect on taste? Nada, apart from hypothetical aging concerns...unless mistreated you should not taste the cork.
A cap is not going to fly off a bottle unless it is not properly seated, no matter the carbonation. It is crimped onto the bottle. The bottle will first explode. It is also incorrect that a cork is necessary for higher carbonated beers. Many of the styles identified at higher carbonation volumes also come in capped bottles (particularly 375ml bottles).
Corks do provide a better seal that slows the flow of air into the beer so those bottles will age more slowly than bottles with caps. That is the primary technical difference between the two.
The primary reasons for corking beer today is marketing. In some areas where corked bottles were traditionally used for bottling it was easier to get champagne bottles or similar bottles that could support carbonation and corking became a matter of convenience as it was cheaper to source bottles locally. However, now it is just a matter of marketing the product. All other corked bottles went into large format, corked bottles to make them look like a classier product than a six pack and legitimize higher prices in the customers' eyes. That is true for the beers you find on the market today in almost all cases. Very few beers go into corked bottles because the brewer truly believes it is necessary for the long term quality of the beer in a way that a cap or a cap plus wax would not provide.
I visited the brewery of West-Malle, Belgium recently.
The bottles of 75 cl have a cork instead of a cap, and it's just because of marketing.