The reason is because the strong force isn't cumulative but the electromagnetic force is. Now, the strong force is a bit more complicated as it does change based on the number of protons and neutrons, but it doesn't build continuously as more protons or neutrons are bound to the nucleus, but the electromagnetic force does.
Say you have a helium atom, 2 protons, 2 neutrons, each is tightly bound by the strong force and the 2 protons are only repelled only by each other. So, it's 1 strong force of attraction, and 1 electromagnetic force of repulsion and the strong force wins. The strong force is 137 times stronger
Now, take Uranium, 92 protons. Each Proton and Neutron is bound to the nucleus by the strong force, but it's only 1 strong force attraction but each proton is now repulsed by 91 other positively charge protons. Hence you have 91 little forces pushing it away. This is much less stable.
Quantum instability always happens at 83 or more protons (1-82 are mostly stable, except for Technetium and Promethium, with 43 and 61 protons respectively), which in and of itself is rather curious. The strong force binds tighter with specific combinations, and as a general rule, even numbers of protons are more often stable than odd numbers. I'm not sure why that is but it seems consistently true.
More on that here: https://en.wikipedia.org/wiki/Even_and_odd_atomic_nuclei and, loosely related, here: http://io9.com/the-oddo-harkins-rule-shows-the-universe-hates-the-odd-1446581327
Also, please see Rob Jeffries answer, as he mentions the weak force - I think his answer is more correct than mine.