For a detailed description, see the lecture notes M. Mustaţă, or by K.S.Kedlaya. For more expository discussions, see Tao's blog or S.W. Park's notes.
For the specialization to one-dimensional curves, chapter 2 of Topics in the Theory of Zeta Functions of Curves by S. Chan might be useful. As far as I can tell, the proof is not explicitly "for curves only", but it is set up so that the one-dimensional case is the intended primary example, worked out in the subsequent chapters.