This has been asked and answered before, but at a level far above anything I could understand.
When proving that: $\Gamma(z+1)=\int_0^\infty x^{z}e^{-x} dx$
is the same as: $\Gamma(z+1)=N^z \frac{1}{z+1} \frac{2}{z+2} ... \frac{N}{z+N}$ where $N \rightarrow \infty$
there is a step that says that we have to use the partial gamma function:
$\Gamma_n(z+1) = \int_0^n t^z (1 - \frac{t}{n})^n dt$
If $\frac{t}{n}$ is replaced with $t$, then the expression becomes
$\Gamma_n(z+1) = n^z\int_0^1 t^z (1 - {t})^n dt$
But I don't understand how this follows.
I'm sure it's pretty simple but I don't know what terminology I should be using when looking in a maths textbook
What you're looking for is called integration by substitution, you can have a look at the wiki article for it: https://en.wikipedia.org/wiki/Integration_by_substitution (it works just as well while integrating a function to $\mathbb{C}$). You should be able to derive the equality from there.