I will explain it as i understand it. its basically an equation. Your shotgun shell has two components that matter to us right now , the propellent and the projectile. Since there are limits on the material that your projectile may be made of manufactures have switched to an affordable material in steel. Now the problem with steel is that its less dense than lead. So to get the energy of the lead loads you need to make steel do one of two things , get heavier or get faster. To make it heavier you increase the size of each projectile ( taking number 4 lead and sub #2 steel). Or you increase the speed , say from 1300 fps of magnum lead loads to 1550 for a magnum steel load.
This all sounds fine , just pick a fast steel load in the appropriate sized shot and your good to go right? Kind of, the problem we now run into is preasure and pattern density ( pellet count). Basically because the steel pellet is less dense ( less weight for the same space taken up) you need to go bigger as previously mentioned. the problem with that is that they take up the same outside space (.15 inch dia for example). so now we are stuffing less pellets into our shell because we had to go to a bigger size for the same per pellet energy. To kill birds effectivley we need a good even pattern of appropriate sized pellets.
The obvious solution is to just shove more pellets in , which leads to the issue of pressure. we are now trying to jam to much stuff into our 2.5 inch space. It compresses on its self and spikes pressure. To reduce preasure we can take some of the powder out which gives more space , less pressure but less velocity as-well. As others have stated , speed kills with steel so taking velocity out is a no-no.
This leaves us with one other option , make the shell longer. this extra quater inch gives us the room to put enough shot in to get a good pattern as well as have enough powder to give us good velocity without hitting spiked pressure ( 3 inch guns i think operate at a higher SAMMI max , but im not sure of that)
so to summarize the 3 inch shell is to give enough room to have both a large enough payload of appropriate sized pellets go fast enough to carry the density and power required to kill birds quickly