You may want to read [Milne, Arithmetic Duality Theorems], Chapter I, Appendix A http://jmilne.org/math/Books/ADTnot.pdf and also [Neukirch, Schmidt, Wingberg], Cohomology of Number Fields, Chapter X, §1.

What makes it easier? The existence of the Jacobian and the known structure of its rational points, that everything is a curve over a field, so the theory is more geometric. Also the Zeta function being rational simplifies things.

Edit: Furthermore, the Chebotarev density theorem for function fields is easier to prove than for number fields (see, e.g., Fried, Function Field Arithmetic, Chapter 6.4 [cf. also http://ricerca.mat.uniroma3.it/dottorato/Chebotarev/JardenChebotarevfunctionfields.pdf] vs. 6.5): In the function field case it follows from the Riemann hypothesis for curves.