What does it mean to sync apps to iPhone? I don't understand exactly what this does, or why I'd want to turn it on. What are the advantages/disadvantages of enabling "Sync Apps" in iTunes to my iPhone?
This Apple Support article provides a quick summary of what syncing does. Basically, as XAlexOwnZX mentions, a sync copies applications, music, and other information between your iTunes account and your iDevice. This process backs up the content (apps, etc.) on your computer.
The advantage of syncing is that if your iPhone gets corrupted, you'll be able to restore it (or its replacement) to working condition, with all of the purchased content that is installed.
The disadvantage is that syncing takes time - however, with iOS 5, you can choose to sync over WiFi and you don't need to tether your iPhone to your computer.
If you Sync an iPhone with iTunes, and that phone gets stolen, you will be able to restore all your app data (high scores, spreadsheets, diaries, databases, etc.) from the iTunes backup to your replacement iPhone.
If not, and the apps don't otherwise store data in iCloud or other cloud storage, all that data will be lost.