In mathematics, a statement and its contrapositive are equivalent. So, for example,
If I bake bread, then the house smells nice.
is equivalent to
If the house doesn't smell nice, then I didn't bake bread.
The contrapositive of the saying from the title of this post is, "That which doesn't make you stronger kills you." When this first occurred to me, I saw it as a funny way to disprove that that which doesn't kill you makes you stronger.
But there is a way in which it's actually true, and almost more motivational than the original. Think about it. What doesn't make you stronger? TV? Donuts? It could be argued that those things do kill you.
If you radically limited your intake and activities to things that at least arguably make you stronger (taking a broad view of "stronger"), you'd probably be pretty bad-ass.