Take the smallest counterexample and delete some member of S and its image in T. This gives a 1-1 function on smaller sets. At the base of the inductive argument, T has one element and S has more, whence the 1-1 function cannot exist.
The name "pigeonhole Principle" was coined when this counting argument was aplied to pigeons, as they occupied a set of protected havens (pigeonholes). When there is an excess of birds, a 1-1 function mapping pigeons to pigeonholes is not possible. Some must remain outside, or share a hole.
Depending on your mood, this concept may seem too obvious to bother proving, yet one should not underestimate its power. Consider an old favorite. Select 9 different lattice points in R3. Prove that one of the 36 segments determined by these 9 points contains another lattice point. Remember that a lattice point is a point with integer coordinates.
A parity argument combined with the pigeonhole principle proves the result. A lattice point in R3 possesses 1 of 8 parity combinations; each coordinate is even or odd. Since 9 points were selected, two have the same parity combination by the pigeonhole principle. Their average, or midpoint,is another lattice point.