For every point to have a solution in $Ax=b$ the matrix must contain at least one set of $m$ linearly independent columns.
But I wonder why for the matrix to have an inverse, we additionally need to ensure that equation $Ax=b$ has at most one solution for each value of $b$. To do so, we need to make certain that the matrix has at most $m$ columns. Otherwise there is more than one way of parameterizing each solution.
I am neither very smart nor very quick to understand in math, don't hesitate to explain it to me with dumb examples as if I was a teenager, I would be very glad.