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).
Very vaguely speaking, the strategy is to bound the height of elements in a set you want to show is finite, say, which lives inside some abelian variety, by looking at the preimage under the canonical map that sends $\mathbb{C}^g$ to your abelian variety. Counting points here requires techniques from transcendence theory and model theory (o-minimal structures), because the things you are looking at have "transcendental parts" which must be carefully dealt with.