There are two sorts of benefits: the benefits to the animals, and the benefits to you. I volunteered for several years as a Dog Volunteer at the SF/SPCA (sfspca.org).
Animal shelters provide many services, the main ones being taking in abandoned animals, providing any necessary medical care for them, feeding them, and helping them to get adopted. They try to provide the greatest care to the greatest number of animals. Because of this, they are often very busy! I helped to socialize, walk and feed the dogs, and help to introduce them to potential adopters. The staff do a wonderful job, but during peak times (such as weekends) there are many more people who want to see animals than staff can accomodate. At times like this, volunteers really help the organization to run smoothly!
The benefits to the individuals are likewise immense. After volunteering for about a year I joined the "Puppy Team" at the SF/SPCA and, literally, got to spend all my time volunteering with puppies. Can you believe this? Most people would pay to spend an afternoon a week playing with the cutest creatures that god created, but I got to do it under the guise of being a "volunteer", a word which most people associate with altruism and sacrifice. Truth be told it was the most selfish thing that I ever did. I loved every minute I spent with those hundreds of cute puppies and my phone is filled with pictures of them and the selfless love that they gave me. (Note: I am told that I actually provided a vital service called "socializiation" - but don't let that fool you - it's really just playing!).
OK, there are more benefits that I received. I started volunteering there because I wanted to adopt a dog of my own, but had no experience with dogs. The SPCA was filled with dog experts (they used to actually have a school to teach dog training, and they run classes to help the public learn to train their own dogs). I learned a lot, and it wound up helping me when I eventually adopted my own dog.
Lastly, I received the benefit of participating in a form of community service that most people just don't have exposure to.