1. During a mathematical education program you will usually encounter it in calculus, for example Taylor's theorem $$ f(x) = \sum_{k=0}^\infty \frac{f^{(n)}(x_0)}{k!}(x-x_0)^k. $$ and the binomial theorem $$ (a+b)^n = \sum_{k=0}^n \binom{n}{k} a^k b^{n-k}, \quad \binom{n}{k} = \frac{n!}{k! (n-k)!} $$ or combinatorics (art of counting). Permutations show up in algebra. On this site my last use of factorials and gamma function was this (at first look rather frightning) equation: \begin{align} \frac{(-n)^{n-1} \Gamma(n+1)}{(1-n)_{n-1}} &=\frac{(-n)^{n-1} n!} {(1-n)(1-n+1)(1-n+2)\cdots -2 \cdot -1} &=\prod_{k=1}^{n-1} \frac{(k+1) n^2}{n^2-kn} \\ &=\frac{2 n^2}{n^2- n}\cdot\frac{3 n^2}{n^2-2 n}\cdot\frac{4 n^2}{n^2-3 n} \cdots \frac{n^3-3n^2}{4n} \cdot \frac{n^3- 2n^2}{3 n}\cdot\frac{n^3- n^2}{2 n}\cdot n^2 \\ &= n^n \end{align} Historically gambling problems were a major reason for the development of combinatorics and probability theory.
2. It is a valid question to extend the factorial, a function with natural numbers as argument, to larger domains, like real or complex numbers. The gamma function also showed up several times as certain integrals, so mathematicians gave it a name and of course noted the relationship to factorials. See the graph at the end of this posting. My favourite application of the gamma function is the volume and surface of a ball in $n$ dimensions: $$ V_n(r) = \frac{\pi^{n/2}}{\Gamma\left(\frac{n}{2}+1\right)}r^n \quad\quad S_n(r) = \frac{\pi^{n/2}}{\Gamma\left(\frac{n}{2}\right)} r^{n-1} $$
3. You ordered that interpolation via "smooth bezier". A Bézier curve is an interpolation function. Drop that part or try different plotting options, see "help plot" within gnuplot. For example:

   plot "factorial" using 1:2 with linespoints

Here is a plot together with the gamma function, or to be more precise, $\Gamma(x+1)$: