Meta Math.SE is truly fascinating to me. Not only that, but I want to contribute, because I believe I can and that I have something to contribute. How can I go about doing this?
If you want to contribute to the math site, you should just go to the main site and find problems and answer them. However, if you really want to contribute to the Meta only, well, from someone who hasn't contributed as much to the Meta as to the main site, my advice is not written on stone, but the least I can say is $-$ you can't force yourself to contribute. The meta is about discussions on how to make the site better. You'd hardly ever see an actual math discussion on the meta site.
If you have questions about the main site, you ask here. That's what people do and if you have answers to contribute that haven't already been posted, you should just answer. But it's not like you see a problem, solve it, look for an alternate solution, and post. If you have more to add to the discussion, just do so. Read the site, learn, and keep an eye open for where you might have something valuable to contribute. But don't go around hunting for places to post.
I agree with all that Ahaan said.
It is great that you want to help out and get more involved on Meta. As Ahaan writes, Meta is the place for discussions about how the site works. It is here that we discuss policies and procedures. We also try to provide help to people who would like to know how the site works. And we can make suggestions for new features.
To participate in theses discussions, one needs primarily to know how the main math site works. So the best first step would be to get involved (by asking and answering questions) on the main site. After you get a feel for how everything works, you will be better suited for participating on Meta.
I suggest that you start reading everything on Meta. This you can do right now. You should, fairly quickly, become familiar with the issues that are being discussed right now. Often people will post links to older Meta posts and you should go read these as well.
You can, also, start voting now. Remember that voting on Meta is different on Meta than on the main site. Don't feel bad about downvoting if you disagree with something. But, also, start upvoting when agree with something. If you don't feel you agree or disagree with something, then just don't vote.
After some time you will probably start to develop your own views and opinions. When you do this, I suggest that you start to comment on posts. Don't comment for the sake of commenting. Comment when you believe that you have something valuable to contribute.
After this, you can start to participate in the discussions. Some people don't like getting involved in discussions. This is up to you. But I believe that participating in the discussions is a good way to help. You and I might agree on a topic, but you might be more articulate than I (most people are!). If/when you do post something just remember that (a) voting doesn't work like on the main site, (b) people will disagree with you, (c) don't loose your temper even when you feel that you are being provoked. Be careful what discussions you throw yourself into. Before posting an answer or a question on Meta research the issue at hand by searching and reading old Meta posts. You should also be aware of the general policies/rules for the main site.
The last suggestion I have for you is to follow the general Meta site for all of Stack Exchange. (This is something that I am not good at).
Here are a list of some key Meta Posts that I suggest you familiarize yourself with:
(I invite others to add to this list.)
I am not a fan of the meta. It's seductive in a way that is subtly but deeply dangerous. It's far easier to introspect and write about the process of, say .. blogging .. than it is to think up, research, and write about an interesting new topic on your blog. Meta-work becomes a reflex, a habit, an addiction, and ultimately a replacement for real productive work. It's something I think everyone should watch out for, whatever walk of life or career you happen to have.