The men doing the catching are referred to as receivers. Not all are eligible to catch passes. In the NFL, running backs must wear numbers 20 to 49, tight ends must wear numbers 80 to 89 (or 40 to 49 if the numbers 80 to 89 have been exhausted), and wide receivers must wear numbers 10 to 19 or 80 to 89. Only those players with appropriate numbers can catch forward passes. This is done in part to help officials distinguish what players can be doing what.
However it's mostly wide outs that are targeted with passes. Athletic ability in the form of speed and dexterity is hugely important for receivers. Speed is necessary to get down the field and past the defense. What makes a good receiver is being able to go and get the ball. Often times, the quarterback will throw the pass away from the defense, forcing the receiver to sprint after it.
This is where dexterity and body control come into play. Rarely will receivers be wide open. Often times they must jump over and among defenders to catch the quarterback's passes. This is why size is so valuable at wide reciver. They don't need to be as athletic if they can just reach up and come down with the ball.
Perhaps the most important reason athletic talent is the ability to stay in bounds. Being aware of your position is the difference between wasting a down and making a valuable catch for the team. This is especially true near the goal line where defenders can easily anticipate where the receivers will try to be.