Q.  Why do the Planets lie in a plane?

A.  They didn't always lie in a plane, they have been brought into one by gravity.

Q.  What is the mechanism?

A.  It's called Equatorial Forcing, A large rotating body causes nearby smaller masses to orbit in its equatorial plane of rotation.

Q.  Does this happen quickly?

A.  No, it's very gradual -- in our Solar System it took around a billion years.

Q.  Does Equatorial Forcing happen elsewhere?

A.  Yes, it's responsible for the shape of the Rings of Saturn. And for the disc shape of our Galaxy, the Milky Way.

Q.  Where can I find out more?

A.  Look at XT806: What is Equatorial Forcing?.