First of all, congratulations joining McK.
This lack of participation is usually linked to 2 points:
1) Lack of knowledge
2) Peer pressure (e.g. introvert)
From your point I guess this is 1) that is affecting you.
Lack of knowledge can again be due to 2 things:
1.1) Lack of prepation
1.2) Lack of basic concepts
Assuming you have as good education as anyone at McK I would say that lack of preparation is affecting your participation level in the meetings.
One way to test this is seeing for example if you are preparing a charter to any problem you are trying to address at your job. Starting by clearly defining your objective function, scope and so on. If you dont have a clear understanding on what problem you are solving, (1) you wont be able to see the drivers afecting the problem, (2) you wont be able to prioritize the most important ones, (4) create hypothesis to those main drivers and finaly (5) see what analyses you need to come up to test those same hypothesis (this in a nutshell is pretty much what you need to do as a consultant).
Ways to solve it I would say 2:
1) Read books on problem solving. I recently read Bullet proof problem solving by 2 former McK partners that I recommend
2) You can do courses that will allow you to speed up. StrategyTV and StrategyTraining will help you. I'm currently watching 90 days consulting that explains the points I addressed here and much more
I'm more than happy to discuss these points 1 on 1 if you want