How do you apply to one?
Normally you don't apply in any formal sense. Ask the project owners or on their internal mailing lists where help is needed. Start implementing a new feature or fix some bugs. If you contribute good work it's a matter of time they'll ask you to officially join the project team.
How do open source projects know
that I am good in what I do?
By looking at your contributions. Good project leaders won't appoint team members before they have contributed something to the project.
How do you find a good open source project from which I can learn?
Simply browse around the major open source hosting sites. Maybe concentrate on the "meta-hosting" sites like Ohloh. The DVCS communities for git (github.com) and mercurial (bitbucket.org) have great hosting platforms that become more and more popular among OSS projects. I do strongly suggest having a look into both, DVCS ecnourages the typical OSS project workflow so much more, you will find it easier to simply start working on something and getting your code reviewed/merged into the mainline.
How open source projects are managed?
Usually there's a project owner/founder and a crew of committers, sometimes with different responsibilities. Well run projects have public/internal wikis and mailing lists. Those are the basics and both are good starting points for you. You'll find there are a lot of differences between small and large projects of course.
How to know if the project is alive?
Simply check the public code repository for the latest commits. Again, meta-sites like Ohloh analyze such things automatically and can give you a good impression of what's going on.