Why does the Wronskian work?

Question

I'm looking at this: https://math.berkeley.edu/~mcivor/math54s11/wronskian.pdf. Which says this: "The fact that the Wronskian is nonzero at $x_0$ means that the square matrix on the left is nonsingular, hence this equation has only the solution $c_1 = c_2 = 0$, so $f$ and $g$ are independent." Where the "square matrix on the left" is the Wronskian and $f$ and $g$ are differentiable functions. I understand how this proves a solution, but I don't understand how this shows that the only solution is $c_1 = c_2 = 0$. Why can't it be any solution as long as $Wc = 0$?