Hearthing to a major city and then flying back to your quest hub can take a lot of time. But you need to do something when your bags are full and you just can't loot that one extra piece of cloth.
One method has gained popularity due to its ease and modularity. Bank alts hold an infinite amount of storage in their mailbox. Technically it's not infinite, as you can only view 50 mails at a time, and each mail can hold 12 items. (50*12=600 storage space possible.)
When you're far away from Dalaran, such as in Grizzly Hills, and you just can't carry anymore bear livers or wolf teeth, you can just visit a mailbox, drop in all the items you want to keep but don't need at that moment, and send the mail to your bank alt! Then you can get back of your way of questing, without ever leaving the zone. Worry about all those items later!
Bank alts should be parked in a major city with a bank and auction house that are near each other. A vendor is also preferred, but unnecessary. It's also very convenient to choose a city other players frequent. Stormwind and Ironforge are the preferred cities for the Alliance. The horde cities are Orgrimmar and Thunder Bluff. With a connection in a major city, your main character can now bind their hearthstone to the current zone they're questing in.
Using an alt to manage your money will also help you save it. Without having your stacks of gold on your main character, you'll be less likely to spend. Easily avoided will be spur-of-the-moment spending. By the time you log on to your bank alt, send the money over, and then log back to your main, you'll have realized you probably don't need to spend 50g for that 1 dps increase.
A bank alt will also let you free up some computer resources. Addons like the Auctioneer package use a lot of computer memory and processing power. When they're enabled and active alongside other addons, your computer can quickly become overwhelmed. When you use a bank alt for managing your auctions, you can disable Auctioneer on your other characters, leaving your precious RAM to be used for more important things.
They can also serve the purpose of taking the burden off logging on your main. Some tanks and healers in large guilds dread logging online due to the shear amount of group invites they'll be expected to accept, even if they only have a few moments of time. Using a bank alt unknown to your guild, you can check your stash of items and gold, and post and purchase auctions unhindered by those guildies you hardly know.
A few people feel the need to equip their bank alt in the best equipment... a haliscan suit and silver cufflinks with a monocle to match. However, many of those items are expensive and you needn't have them to utilize the bonuses of an alt. You simply need a snappy name (iamabankalt, dontrenameme, sirbanksalot, iliketobank, richrthanaig) to begin the trade.
Update: I've thought of a few more reasons you should have a bank alt.
Having multiple bank alts lets you sort items conveniently. Zul'Gurub and the Ruins of Ahn'Qiraj have zone- and faction-specific items that are only useful after saving up for a certain time. Now that those two instances are easily farmable on a regular basis, these items can become bulky on a main and take up your entire bank if you're not careful. Making an alt (named "zgfarmbank" maybe?) can be a good way to store those items until you have enough to use.
Placing and hawking items anonymously can find you prospective customers you might not have found on your main. If you've grouped with players who turned out not to like you, or your guild is infamous, people might be wary of your items and would not buy from you, even if you're offering the cheapest wares. Using a bank alt lets everyone buying from you know that you're an alt, clearly, but no one will know who your main is (as long as your have a dissimilar name, and aren't in your main's guild). And trust me: No one ever asks. Generally, no one cares because most people use or know people who use bank alts.
Having a handy bank alt will save you trouble! Sometimes I don't feel like going to town in the middle of a good grind/quest session to post my auctions and put up with trade chat. Other times, I'm in a group and don't want to abandon them or burden them to wait on me. With a bank alt, you can put off handling the tedious details of trading until you're more fit for the task. If you're not in the mood for it, it shouldn't be getting in your way until you do it.
And lets not forget the roleplaying opportunities. With a properly-equipped alt, you can partake in the major-city roleplay scenes quite seriously, with a very humorous character. Having a class you wouldn't like to roleplay as a bank alt can let you experience their character when you're in the mood, and keep to your main the rest of the time. Also, since no one knows the alt is you, you can have fun with comical gnome or undead characters, or a stuck-up blood elf. Just remember to respest your fellow roleplayers and the roleplay-realm rules, as always.