What are the rules for factorial manipulation?

Ignore the $-1$, since they occur on both sides. Then you have:

$$(k+2)! = (k+2)(k+1)! = (1+(k+1))(k+1)! = (k+1)! + (k+1)(k+1)!$$

Basically, it's just the distributive law.

Assuming we are given only the task:

Simplify the following: $\quad (k+1)! - 1 + (k+1)(k+1)!$

Let $\color{blue}{\bf x = k+1}$. Then we have $$ \begin{align} x! - 1 + x(x!) & = x\;(x!) + 1\cdot x! -1 \\ \\ & = (x + 1)\;x! - 1 \quad\quad\quad\quad\quad\quad\tag{distributive property over addition} \\ \\ & = (\color{blue}{\bf x} + 1)! - 1 \\ \\ & = (\color{blue}{\bf k+1} + 1)! - 1 \\ \\ & = (k+2)! - 1 \end{align} $$

Collect together the two terms that have a factor of $(k+1)!$:

$$\begin{align*} (k+1)!-1+(k+1)(k+1)!&=\Big(1+(k+1)\Big)(k+1)!-1\\ &=(k+2)(k+1)!-1\\ &=(k+2)!-1\;. \end{align*}$$