There are many answers to this question as I'm sure you're aware but the most basic answer is as follows:
Einstein's theory of special relativity dictates that an object gains mass as it accelerates. The faster an object moves, the more mass it gains. The more massive it becomes, the more thrust (Energy) it requires in order to maintain its speed. According to special relativity, if an object with mass were to achieve the speed of light, its mass would become infinite. Furthermore assuming that the object did achieve infinite mass during light-speed travel, to keep moving, the power (or Energy) behind its thrust would need to be infinite and as no force in the known universe can achieve this, light speed travel is impossible.
See also: E = mc^2 (E = energy, m = mass & c - speed of light for mass/energy equivalence.