Fly fishing is a contemplative sport that is not quite the same as regular, or spinning reel, fishing. You cast a line in both, and you wait for fish to bite in both. But fly fishing's reliance on the rod and line to create a good arc for the fly bait, compared to spinning reel's weighted baits doing the work, means that even experienced fishers have a learning curve when they switch to fly fishing. If you want to try fly fishing, especially if you aren't that experienced with fishing to begin with, you can hire a guide, or you can take classes.
If you're the type that likes to ask questions—a lot of questions—a private guide is a better choice for you. You'll get one-on-one teaching and don't have to worry about hogging class time with your questions. Guides can also give you information that is more specific to your situation. For example, if you're a complete beginner, you can discuss the bare minimum supplies you'd need to start fishing at a local city lake. If you're already an experienced fisher with spinning reels, the guide can skip over a lot of the basics and concentrate on the differences between spinning reel fishing and fly fishing.
Finding a Community
Classes offer the benefit of finding a community. When you learn to fly fish with a guide, you get great advice, but the guide won't act as your fishing partner. You could hire the guide again and again for additional lessons, but that's all. In a class, you and your classmates will eventually form little groups, much like you and your friends did in high school and college classes. Those little groups can become your fishing buddies.
Costs and Equipment
Classes can be a great way for people who don't have a lot of financial resources to get into the game because these classes sometimes have equipment you can rent or borrow. If you're not sure you'd like fly fishing, for example, a class where you can temporarily use equipment is definitely the best option, so you don't spend money buying equipment you won't use again.
Of course, you can do both, too. Maybe you can take a class initially to get your feet wet (no pun intended, really) and then work with a guide to refine your skills. Or you could hire a guide to learn the basics and then take classes to practice your casting technique and find fishing buddies. Both are options and can help you get better at the wonderful hobby of fly fishing.