If the functions $f_i$ are linearly dependent, then so are the columns of the Wronskian as differentiation is a linear operation, so the Wronskian vanishes. Thus, the Wronskian can be used to show that a set of differentiable functions is linearly independent on an interval by showing that it does not vanish identically.
Here's what I don't understand. If $f_i$ are dependent on some $I\subset \mathbb R$, then there is a point $x_0\in I$ at which there is a non-trivial linear dependence relation (because linear independence means that any relation is trivial for all $x\in I$). Then the Wronskian vanishes at the point $x_0$. The negation of this would be: if the Wronskian does not vanish at any point in $I$, then the $f_i$ are independent on $I$. But that's not what Wikipedia says. (It says if there is a point $x_1\in I$ at which the Wronskian is non-zero, then the $f_i$ are independent.) Am I wrong?
Also, at what point does the proof of the converse fail?