Tools from transcendence theory have been crucial to the most significant recent advances in problems of unlikely intersections. The strategy was first dreamed up by Zannier, I believe, and has been applied with great success by Pila and Habegger, for example here: http://arxiv.org/abs/1409.0771 The authors prove some major cases of the the Zilber-Pink conjecture, some of them unconditionally, some conditional on conjectures in transcendence theory. For people who don't know, the Z-P conjecture is an extremely general and very strong statement in diophantine geometry. As an example of it's strength, Pink has given a very short argument that reproves Faltings's Theorem assuming only a very special case of the Z-P Conjecture.
In Zannier's monograph Some Problems of Unlikely Intersections in Arithmetic and Geometry one can find a great introduction to these problems, as well as some indications of the applications from transcendence theory. Transcendence techniques are also discussed in some of the appendices (by David Masser).