We all know functions of the form $ax+b$ are easy to invert. While reading about the inverse function theorem this made me think the following,
"If we have a $C^{1}$ function then we can linearize it locally and this line which essentially is of the form $a+bx$ is locally bijective"
Is this a good intuitive way to think about it?
I would say your interpretation is fine for the one dimensional case. The way I like to think about this is by imagining a linear system of equations, just as if my function $f(x_1,\cdots,x_k) $ is given by
$$f(x_1,\cdots,x_k) = A\textbf{x}$$
For some matrix $A $.
It is essentially a generalization of your thoughts.
Well, if your function $f$ is sufficiently well behaved in the neighbourhood of a point, then you can think of the function as being essentially a linear function of $\textbf{x}$. And if it so happens that $\det{A} \neq 0$, then the matrix is invertible and you can recover the coordinates from the value of the function:
$$f(x_1,\cdots,x_k) = \textbf y \iff A\textbf {x} = \textbf{y} \iff \textbf{x} = A^{-1}\textbf{y} $$
That is why, in particular, you need to ensure the jacobian is different from 0, so you can think of inverting this matrix.