In this context, it is similar to saying that their roots have not quite grown yet, so they may yet fall out. It can contain the idea of something small being accepted or incorporated into a larger thing and having the desired effect. For that reason this use of the verb is used for things like organ transplants, implants, or similar things like that. 'Take' has a lot of uses.

It means "taken root" or "taken hold". If Rachel moves Barry's new hair too roughly, it will just fall out. Do a search on Barry's line, "Careful! They haven't quite taken yet." This scene has been discussed on other language forums as well.

finished done or completed