Let f(x) = y. The guards have f and y. A spy gives a guard an x, and the guard computes f(x) and checks it against his list of ys. If it's on the list, the spy can pass. The enemy has f and y too, but it doesn't help them! f(x) is trivial to compute but the inverse is hard, so the enemy knows exactly the answer they want the guard to get and no idea how to make him get it!

One-way functions are very cool and form the basis of public-key cryptography, although it's quite a bit more complicated than this example.