You've covered what time dilation is (or might be) but I think you've missed what makes the "paradox" get called a "paradox" in the first place.
Originally Posted by Durakken
...which is that each observers view is equally valid.
(Bob in the spaceship can view himself as standing still, and Alice, sitting on Earth as zooming away then coming back again. Bob's view is entirely valid, because there is no absolute reference frame which which to say it's Alice or Bob who is moving "faster". So why doesn't Alice, on "spaceship Earth", come back younger? While there is relative motion between them - they each see each others clock as running slower than their own. It makes no difference who is "faster" or "really moving".)
The twins paradox is not about time dilation, as such, but about the apparent lack of symmetry - given that all points of view are equal.
The answer (as I understand it) is not that Bob moves "faster" but that Bobs' reference frame changes, and as a result he moves further through space-time.
(While Bob in a spaceship uses acceleration to change frames, messenger clocks synchronised in some way, don't need to invoke acceleration, to shift reference frames.)
Get up, a get-get, get down.