John is right on.

Just to add one bit. You might be wondering why trainers do this, why not just adjust the wing and tail incidence to not require down thrust.

The reason is that for a trainer, it's desireable for the plane to try to pull itself out of a dive. In a dive, the engine thrust doesn't change (much), but the airspeed increases. The angle of the wing plus the higher airspeed in the dive causes an increase in lift, and the plane to lifts the nose and pulls out of the dive.

But this same effect causes the nose to really go up in a hurry when you add power, so, as John said, downthrust is used to reduce the amount the nose goes up when power is increased, but still keep the tendancy to pull out of a dive when power is not changed.