1. You don't have time to teleport back to Azeroth and use the Auction House. Let's face it, you're a busy person. You're working on dailies, you're heading into Heroics, you're going this way and that, and you really can't afford to blow your Hearthstone cooldown on a side trip to a capital city. This is where a bank alt comes in. You just send all your stuff to them, switch over, and throw the stuff on the AH without missing a beat on your main. Also, If you have multiple stacks of some stuff (As I often do with Netherweave Cloth, Arcane Dust, and Greater Planar Essences), it's nice to be able to store them so you don't flood the AH and bring down prices.
2. You need more bank and bag space. If you're a raider, a hybrid, or a packrat, chances are you're pretty low on bank and bag space. You need multiple gear sets for DPS, Tanking, Shadow Resistance, and whatever other things pop up, plus, you'll want to keep around a few cool looking quest items or old gear sets just because they look awesome. If you're that type, chances are your bank and bags are both filled to bursting, you just don't have space for anything else. Being able to mail all your extra stuff to a bank alt can free up space quicker than you might think.
This even works in the middle of a play session. When you're loaded down with Motes of Mana, Arcane Tomes, Sunfury Signets, and various green items from doing the outland daily circle, just stop by the nearest mailbox, send them to your alt, and be ready to out for another round of farming. If you want to be a truly hardcore bank alter, you can even create a new guild specifically for your bank alt, and have full run of an extra guild bank tab to store more of your stuff.
3. It's nice to get a little more centralized. If you have 2 or 3 alts with different professions all using various cooldowns and combines to make stuff to sell and use, it can get a bit annoying to keep track of all of it. Yes, there are addons you can use for that stuff, but it's also nice to be able to know that you can go to one paticular character and always find what you need, and send it off to the character that needs it instantly.
I admit, sometimes I sort of get annoyed that I have to keep a slot in use for my bank alt when there's always that one extra alt I've been thinking of making, but in the long run, having a bank alt has been nothing but awesome for me. It's great little time and space management tool, and if you've been on the fence about making one, I'd recommend that you give it a try, especially if you're a busy 70 with bag space problems. You'll be glad you did.