What benefits do bike clubs provide?
I'm going to disagree with Frisbee and note that there are many cycling clubs which aren't race teams but are just a bunch of people riding together and supporting a community of cycling in the area. 

There are several things that you can get:


Group ride organization (support for the rides, sometimes food, planning, etc.)
Discounts at local bike shops and for certain group rides which cost money
Access to a workspace (shared tools)
Meeting people who bike in your area (social networking too, like Strava groups or whatever)
Advocacy for local cycling efforts (trails, etc.)
In some cases, the clubs run race teams as well (which will increase costs, to get team kit and stuff). 
Insurance (as pointed out by RiderX in the comments)


That being said, is it worth it? Look at the clubs in your area and look at how much they're charging. If it makes sense to you, you may want to go for it. If not, don't. A lot of clubs are only 15-20 dollars a year, some will be good, some will be bad. Some will be more expensive, and some will be free, depending on how much they're providing and what not. 

The best thing to figure out to avoid or join a club is to talk to cyclists in your area -- you may want to visit bike shops you trust and talk to the staff there, and other cyclists you know in the area. If you're new in the area, you may have some work to do in this regard.