I often find myself fermenting under pressure as I try to keep my kegs flowing and my wife happy, but that is a different kind of pressure fermenting!
It is likely you have already heard and read plenty about this fermentation technique, so I will endeavor to just offer a review of the pro's and con's instead of a technical deep dive.
Given the availability of fermenters that are designed to withstand pressure up to about 35PSI, including Corny kegs, plus a range of inexpensive spunding valves, fermenting with pressure is remarkably easy to do, but some still ask why we would bother.
At temperatures in the 70 - 75F range, many yeasts will produce esters (and fusel alcohols) that will be evident in the finished beer. This is normal behavior for yeast. If this flavour is desired, such as in a Hefeweizen, there is no advantage in pressurized fermenting. However, if you are aiming for a lager or ale that does not exhibit esters, applying pressure during the active fermentation phase will suppress or eliminate ester production. For homebrewers without the ability to control their fermentation profile by means of a fermentation chamber or other refrigerated methods, fermenting without esters in the low to mid 70's F is easy, sometimes as easy as just resting the fermenter in the kitchen. Since yeast is more active at these temperatures, the overall "grain to glass" time is also reduced. While this is an advantage for some, there are many homebrewers who will not be looking to shorten the overall fermentation time as much as they are striving for clean(er) beer in more common areas of their homes that are not temperature controlled. For those hell bent on a 7-day grain to glass bevy, I think Kveik yeast is more the answer than pressure fermenting.
One of the downsides of fermenting under pressure is the impact the pressure has on the physiology of the yeast cell. It both hinders replication (growth) and viability. This can mean slower attenuation and possibly even stalling, which should be correctable by reducing the applied pressure via venting and rousing the yeast. This impact may also mean a lower ability to "clean up" the beer following primary fermentation, again correctable by reducing the pressure. This is why you might read suggestions to ferment at a higher pressure until FG is reached, then reduce the pressure for the duration and relieve the stress on the yeast. Spunding valves make this a very simple process to control.
Another advantage of fermenting under pressure is the possibility to utilize the evolved CO2 within the vessel to self-carbonate the beer. This will require you to retain much of the CO2 within the fermenter and then cold crash the fermenter to dissolve it into the beer. The result is carbonated beer directly from the fermenter, although I hear the carbonation level sometimes needs to be boosted with additional external CO2. No matter how much you need to bump the carbonation up with, you are still saving CO2 in the process, and thus saving costs. If you bottle your beer, bottling carbonated beer (as you likely know only too well) if more difficult than flat beer, so keep that in mind.
I have been asked if there are yeasts that specifically perform better than others under pressure, and I have not been able to answer this from personal experience, yet. Many people are using lager yeasts (Lallemand Diamond, Fermentis W-34/70 or S-23, or Escarpment Labs Isar or Mexican Lager) with wonderful clean results, but also using most (if not all) of the ale strains for clean results with no off flavours. Pressure fermenting with Kveik yeast, on the other hand, is not as well reported. I suspect that the relatively low temperatures talked about here would essentially leave the Kveik more or less dormant, since it thrives at temps much higher. However, if you were fermenting in the 85 - 95F range with Kveik and wanted to promote a cleaner result, pressure might still do the trick. More on this to come...
So, in closing, I would say that fermenting with pressure is another exciting tool for homebrewers, and one that may eliminate the frustration experienced by small-space brewers who don't have a temperature controlled chamber for lagers. With a fermenter capable of safely holding the required pressures, you now have the option of fermenting with pressure, or without, but with many of the more common fermenters (glass carboys, pails, most plastic fermenters) there is no pressure option.