The inverse of a function does the opposite of the original function. If the function takes an input $a$ and produces an output $b$, the inverse function will take $b$ and produces $a$: $$ \begin{equation}\begin{aligned} f(a)&=b\\ f^{-1}(b)&=a\\ \end{aligned}\end{equation} $$
Only bijective functions (those which are both injective and surjective) have inverses.
The inverse of the inverse is the original function
Consider the functions $f(x)$ and $$ \begin{equation}\begin{aligned} g(x)=f^{-1}(x)\\ \end{aligned}\end{equation} $$
Then the function $$ \begin{equation}\begin{aligned} g^{-1}(x)&=(f^{-1})^{-1}(x)\\ &=f(x)\\ \end{aligned}\end{equation} $$
Example
Given that $f(x)=\frac{1}{x+3}$, find the function $f^{-1}(x)$ and its inverse. Let $f(x)$ be $y$: $$ \begin{equation}\begin{aligned} y=\frac{1}{x+3}\\ \end{aligned}\end{equation} $$
Interchanging $x$ and $y$, and solving for $y$: $$ \begin{equation}\begin{aligned} y&=\frac{1}{x+3}\\ x&=\frac{1}{y+3}\\ x(y+3)&=1\\ y+3&=\frac{1}{x}\\ y&=\frac{1}{x}-3\\ f^{-1}(x)&=\frac{1}{x}-3\\ \end{aligned}\end{equation} $$
Let $f^{-1}(x)$ be $g(x)$: $$ \begin{equation}\begin{aligned} g(x) &= f^{-1}(x)\\ &=\frac{1}{x}-3\\ \end{aligned}\end{equation} $$
Thus by finding the inverse of $g(x)$, we will find the inverse of $f^{-1}(x)$. Let $y$ be $g(x)$ and interchange $x$ and $y$, and solving for $y$: $$ \begin{equation}\begin{aligned} y&=\frac{1}{x}-3\\ x&=\frac{1}{y}-3\\ x+3&=\frac{1}{y}\\ (x+3)y&=1\\ y&=\frac{1}{x+3}\\ g^{-1}(x)&=\frac{1}{x+3}\\ \end{aligned}\end{equation} $$
Thus: $$ \begin{equation}\begin{aligned} (f^{-1})^{-1}(x)&=g^{-1}(x)\\ &=\frac{1}{x+3}\\ &=f(x)\\ \therefore (f^{-1})^{-1}(x)&=f(x)\\ \end{aligned}\end{equation} $$
Hence the inverse of the inverse of a function is the original function.