What Havoc said.
DSLRs are fairly easy to learn, but can take a lifetime to master. They won't magically improve the quality of your photos. That comes with time and practice. You can get absolutely stunning photos with a P&S, and you can get utterly horrid photos with a $3600 DSLR. 90%+ of the image is in the photographer.
There are a few good reasons for the hobbyist to invest in a DSLR, though:
When it comes to scalability, DSLRs are without peer. You can get lenses, filters, flashes, remote shutter releases, different mounting apparatuses, etc. Of course, all this costs.
Lenses require more attention here: the photo comes through the lens, so it is arguably the most important part of your camera. Interchangeable lenses give you so much more flexibility and creative range. There are some smaller camera systems out there with interchangeable lenses which will do 90% of what most people need, but the best glass is with the DSLRs.
Modern DSLRs are capable of doubling as HD camcorders, and they do a pretty darned good job of it. P&S cameras can also shoot video, but DSLRs are increasingly becoming high-quality video specialists. Some TV shows (such as House) are using higher-end DSLRs as video cameras because they're small, lightweight, take professional-quality video and offer features dedicated video cameras cannot (particularly regarding depth of field).
DSLRs grant a much finer level of control over the nuances of a photo, although better P&S cameras can come close. By that, I mean that you can use manual focus with a DSLR, and it's easier for you to make decisions regarding depth of field or very high shutter speeds or what have you. But this takes time and practice.
If you're simply trying to take better pictures, a good P&S and an understanding of photography will serve you well (the book "Understanding Exposure" comes highly recommended.) If there are specific features that DSLRs offer that other camera types do not, then they are absolutely good values and worth the investment. I would also suggest that if you are seriously looking into photography as a hobby you want to invest in, take the plunge. It's better to learn to use good equipment than it is to buy something you're going to replace in a year or 2, and then re-learn anyway.