Obviously, clothing that dries quickly makes life easier if it accidentally gets wet from tropical rainstorms/saltwater spray from friendly dolphins/spilled Oktoberfest beer. Wet clothing that stays wet is uncomfortable and sometimes embarrassing. Remember, Murphey's law states that all spilled liquids will magically aim for the crotch of your pants. The longer it's wet, the more stupid jokes you have to make up about your 'little accident.” More importantly, packing wet clothing in you backpack causes problems, including making half of what's in your bag smell awful for the rest of the trip. Actually, that's more of a symptom. I'll let you imagine the problems it indicates for yourself.
The other big benefit of clothing that dries fast is that you can do your own laundry and have it ready to go soon later. Saves time and money over the ho(s)tel's laundry service, even if it “only takes 24 hours.” Since you won't be carrying that much clothing, you're going to need to do laundry a lot more often than you do at home. You'll want a drain plug, some laundry detergent, and a clothesline.
One thing I should clarify: when I say “quick dry,” I don't mean it has to be specialty clothing with a label somewhere that says Quick-Dry(tm). It just has to be made of a material that dries fast. Just take a look at the laundry label. Synthetic materials like polyester are usually the easiest of these to find. Nylon is also good, most commonly found when looking for pants. The material you want to avoid is cotton. Unless it's specially treated, cotton almost always takes forever to dry.
Unfortunately, cotton also what about 90% of clothing is made of today. So we need to find that other 10% I tend to look for sports clothing stores, outdoor supply stores, and occasionally, if I find them, dedicated travel supply stores (tough to find outside the US). Then of course there's the internet (don't forget overstock sites for discounts and googling for coupons before checkout).
An extra note on the specialty travel stores. They are good, but they're often the most expensive option, and you run the risk of walking out with an outfit more suited to a colonial era safari rather than modern Europe, Asia, or anywhere else today. Buy stuff you wouldn't get odd looks for wearing at home.
Now, one thing to note is that even the most expensive SuperDooperUltraQuik-DryMax shirt from BiggAdventure inc. isn't going to dry immediately in cold or damp places if you just ball it up and throw it in a corner. Drying this stuff out is going to take time. If it's completely soaked, it'll sometimes take a few hours. If you're smart, it'll take fewer. Here are three ways to dry your stuff out faster:
1) After wringing it out, roll it up in a dry towel and squeeze. The towel will take a good chunk of the moisture out. Yes, you will then need to dry out the towel, but since you're using it after you (I hope) shower, you had to do that anyway.
2) Hang it up, with space on either side of it, in a place that is warm, airy, and dry. In that order. In other words, hanging it next to your bunk in a heated dorm room is better than on a line outside in near freezing temperatures, even if it is airy and dry out there.
3) Wear it. Given a little time, your body heat actually dries damp clothing out pretty well, as long as you're not wearing much over it to keep the moisture in.
And there you have it! Clean clothing, and a bag that doesn't smell (much). Enjoy.