Really you don't do analytic continuation to extend the factorial fuction to the gamma function. Analytic continuation is when you have a function defined over a region, like a whole interval of the real line, not a discrete set of points (even if an infinite set) like the factorial function.
Instead, to extend from a discrete function to a continuous function, the more appropriate terminology is to call it "interpolation". Once you have the gamma function that "interpolates" the factorial function to $[1,\infty)\subseteq\Bbb R$, then you can use analytic continuation to extend it to (most of) the rest of $\Bbb C$.
Anayltic continuation can be understood by what happens with polynomials. Two polynomials that are equal on an interval are equal everywhere. Functions that can be expanded in power series (which are like infinite degree polynomials) have the same uniqueness property. That uniqueness property allows us to extend them uniquely (if at all).