Different groups of people have different ways of communication. In some groups, everyone is expected to not speak while someone else is speaking, instead waiting until there's a break where it's easy to insert oneself. In others, everyone is expected to interject and break in while the previous person is speaking, as long as you're keeping on the same track. In others still, interrupting and taking over the conversation is just how everyone does it, and if you wait for a "good moment" you will never get a word in.
None of these ways are wrong in themselves, but if you're used to one style, it can be hard to adapt to a different one when you get into a new group. It seems to me that you're most comfortable with the first variety, and your coworkers are of the second or third one. It wold be nice if the boss/facilitator/whoever is in charge of the meeting could make sure that everyone gets heard, but they may not even be aware that there is a difficulty. (If/when you are the leader in a group or meeting, remember this and try to do better!)
How to interrupt
I'll answer the second question first - how to gracefully get in there. My advice is to look at how other people do it. Do they interrupt? Do they make gestures to get everyone's attention? Do they lean in and go "uh-huh, yeah, also I think that..."? There will be some sort of unwritten rule or standard here - you need to figure out what that is. Once you've done so, you can apply that rule to yourself - if it's OK for them, then that means it's OK for you.
How to get the confidence to speak
The first question is harder. It's easy to say that you need more self-confidence, but it's hard to get there... Is there anyone in the group that you feel comfortable with? If so, could you speak to them after a meeting? Think of a point that you wanted to make during the meeting and bring it up with them - something like "Oh, after the meeting I got to thinking about the screwdrivers - what do you think about reversing the polarity?" You'll get feedback on your idea without it being in front of a dozen people. Also, you're building a rapport and a reputation - next time you're in a meeting, they will be supportive from the start because they already have a good opinion of your ideas.
At this point, every time you go to a meeting and don't say something, it will be harder to say something at the next one. You're building it up into a huge enormous decision. The only way to get past is through - make up your mind to say just one single thing. It doesn't have to be brilliant, it just has to be one single thing that you said. If you manage that, you have won. The one single thing can be as short as "I think that what Jim said is very interesting." At this point, your goal is to break the silence.
Next time, go for two things. After a few meetings when you've actually spoken, the act of speaking will be less of a hurdle. Then you can start thinking about content - about getting your own ideas out. You'll be used to the conversational style and be able to focus on your thoughts rather than on the situation.
You should also remember that most people will not even be aware that you are uncomfortable with speaking. They will be too busy wondering if what they themselves are saying is smart or stupid, or if they missed something that's obvious to everyone else, or if they should maybe just have shut up. You just can't see it by looking at them - and neither can they, by looking at you. They will spend so much energy worrying over what impression they themselves are giving that they have very little left to spend on judging you.