From the reference frame of the garage, the ladder is moving with a relativistic velocity and therefore experiences length contraction. From the equally valid reference frame of the ladder, it is the garage that is moving with a relativistic velocity, making it even smaller (so the ladder wouldn't fit).

There's a paradox so long as you're still thinking time is some sort of constant and not just another compressible dimension like the others. Its hard to do this at first. Most people end up with a half-muddled "time 'goes slower' when" without fully getting the implications of time contracting as well. The ladder "paradox" is designed to illustrate this to beginners.