To keep the brewing stand hot, so that it can melt and mix the ingredients. As you know, you need a blaze rod to craft one and blaze rods are very hot. But after a while, those rods will just cool down. So to keep them hot and thus keep the brewing stand working, you must feed it new blaze powder every so often.
I'm not actually complaining, I'm just wondering why. Minecraft is a fairly complex piece of software, do we make changes to it over nothing? I'm thinking there *must* be a reason, possibly a good reason, and I'm curious as to what it is.
There really isn't much of a gameplay use for it. I think it's mainly for balance and to make it slightly more challenging. That way you have to at least go to the nether sometimes after your first few trips. If you have a blaze farm though, it's not much different.
Jeb said that he wanted progress to be challenging in Minecraft, and as brewing stands can be found in igloo basements, people could brew potions without ever killing blazes. As the only way to get blaze powder is via blaze rod (witches used to drop it but that was removed), players have to kill blazes to brew potions.