In solving differential equations, we see vector spaces inside the set of all functions all the time, and its important to be able to see whether a set of functions is linearly independent or not.
Linear independence is defined for functions the same way it is for any vector space.
The way to prove this statement is to understand what it would mean to be false. If that statement were not true, then the set could be linearly dependent, and the Wronskian might be nonzero somewhere. Take the transpose to make it feel better, and do row-operations on that transpose. So, the determinant would be 0. The other direction is stickier.