It's hard processing what happens in meetings - How can I be more effective?

I don't want to seem dumb...

Flip this mentality! There's nothing wrong with seeming dumb if you quickly improve. Embracing being "dumb" means that you will grow out of that state much quicker. Of course, you don't want to derail the meeting by asking 15 questions or something, but I definitely think 2-3 should be okay. If you're new and you have a good team, people will definitely understand.

In terms of more concrete tactics, I heavily recommend this Q&A about being more active in meetings. In particular, try to "do homework" on the meeting beforehand, and the meeting invite doesn't have a clear agenda + relevant materials, don't be afraid to ask the meeting owner for them!

Lastly, there are a lot of tactics you can use to alleviate the anxiety around asking questions and making your team far more accepting of them, even if they're "dumb". For that, check out this Q&A. A big Meta-specific hack is prefacing your question with "n00b question" - You can even use this if you aren't new to the company in situations like being new to a tech stack or when you have a suspicion that the answer to your question is obvious.