The Van Allen radiation belt is a region in space, surrounding the earth.

Read the whole article of the first finding of Van Allen

And to get through to the moon, you will have to pass through them. And the radiation is deadly, at a 2500 rem (where 500 is deadly) you will die, if you stay long enough and were not shielded.
One thing though about the type of radiation that occurs in the inner belt (with the highest amount)

So how did they go through alive:

  • There was no solar particle event
  • They spend only 15 minutes in the highest risk zone
  • The command module was heavily shielded
  • The route they took went through the outer regions of the belt.
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Sources :  

Apollo Experience Report - Protection Against Radiation; specifying what was done as measures for safety on the apollo program show that the risk at the time was low, because there was no major solar-particle event. Also the command module was heavily shielded and the crewmen would have recieved 60 to 100 rads to their skin and 10 to 30 rads to their blood forming organs.

The Deadly Van Allen Belts : they took as little time as possible to go through the belt. They spend only 15 minutes in the most dangerous zone. In total they would have received only 6,5 rads/hour, where 300 is lethal.

So why is it still a problem?
There are actually two problems:

  1. The instruments on board (computers) are more fragile than they were in the '60s and '70s. And they do much more of the work than in that time.
  2. The astronauts have to spend more time in space, because we are not going to the moon, but to Mars.

A video if the new Orion mission explains a few of the obstacles,

NASA deep space exploration: Jeffrey Rusick at TEDxElonUniversity talks about the difficulties of space travel.
This is the problem they are trying to solve, not short missions to the moon, but the longer missions to mars.