I'll be brief -

After the new teaser for Toy Story 4 showing a new toy made out of a spork and some household objects claiming that it didn't belong there and that it isn't a toy, I think we can now determine what it is that brings toys to life.

It's a simple answer. The power of the imagination, in children, has some sort of magical quality that gives toys the ability to come to life. Presumably, this is permanent, and not something that just goes away once children grow older.

As we see in the teaser, this means ANYTHING can be a toy so long as a child wills it so via their imagination.

Bonus: This implies Cars CAN and maybe DOES take place in the same universe as all the other Pixar films, in a possible post-apocalyptic future where Cars were brought to life via the same method, but then eventually became self-sustaining post-humanity.