Simple. Insulin increases the activity of the famous (and nearly cellularly ubiquitous) sodium-potassium pump. You know - the one that sits in the cell membrane and tirelessly brings 2 potassium atoms into the cell at the same time as kicking 3 sodium atoms out?
Anyway, that pump obviously has the general tendency to decrease the blood potassium level (since the potassium will leave it to enter a cell), and so increasing the activity of the pump is one way to treat hyperkalaemia (elevated potassium levels).
This is what insulin does. High serum potassium actually stimulates insulin secretion by the β cells of the pancreas. And the insulin forces potassium into cells by means of up-regulating the above pump.
Incidentally, β2 agonists (like salbutamol) seem to accomplish the same feat via the same mechanism.