The Van Allen Belts were discovered in the late 1950's at the dawn of the Space Age when geiger counters were put on satellites. These instruments detected high energy particles. Since then, we have learned that there are two main donut- shaped clouds, and a third weaker cloud which comes and goes depending on the level of solar activity. The inner belt contains high-energy protons and is a severe radiation hazard to astronauts working in earth-orbit. Because of the shape of the magnetic field, parts of this inner belt dip down to only a few hundred miles above the earth, especially over the region near Brazil in the so-called 'South Atlantic Anomaly'.
The belts are not important to life on the surface of the Earth, but you can turn this around and say that, if there were no belts, the earth would not have a magnetic field. This means that cosmic ray particles would be free to collide with the earth's atmosphere in greater numbers, and from these you would get a higher background level of secondary 'air shower neutrons' which would lead to higher background radiation doses on the surface. The Belts do not directly impact human life, but if they were gone, this would imply that other effects would have to be in effect as a consequence of what the larger system would look like.