I think that references in this field can be of interest and useful also at distance of time from the original question.
Transversality conditions are a complex subject in calculus of variations and optimal control theory.
From a (mathematical) economics point of view, classical references are, of course, the books of Takayma, Analitycal Methods in Economics, and Mathematical Economics, where optimal control and calculus of variations are exposed in detail, of course with applications to economics.
If you want a deeper insight from a mathematical point of view, there are a lot of books in calculus of variations and optimal control theory.
A classical reference in calculus of variations is Gelfand and Fomin, Calculus of Variations, Dover Publications Inc., 1991.
A good introduction to optimal control theory is Macki-Strauss, Introduction to Optimal Control Theory, Springer-Verlag, 1982.
For optimal control theory, if you have the curiosity of seeing original sources, there is the original text by Pontryagin, L.S. Pontryagin, V.G. Boltayanskii, R.V. Gamkrelidze, E.F. Mishchenko, The mathematical Theory of Optimal Processes, Wiley (1962), where the 'Pontryagin's Principles of Maximum 'are stated.
Actually, the 'Principle of Maximum', is not a single theorem, but a collection of theorems, and the question of transversality conditions is analyzed in each case.
But I warn you: the proof of the first Maximum Theorem in Pontryagin's book is about forty (40) pages!
That's to say that it is not a simple subject.