Dave MacLeod explains body tension here.
“Body tension is the product largely of technique, but also of strength through the body. Some important (and trainable) parts of it are:
Climbing rhythm. That is not getting too extended with both arms high before moving the feet.
Aggression in the lower body. Many climbers are far too passive with the lower body, and aren’t using the strength they already have.
Placement of the foot - The big toe must be in a position to apply the strength, and it’s often not possible because the toes are not engaged and the heel is dropped low.
Turning of the trunk - Helps bring the tensioned hip close to the wall during a stretch and be more ‘over’ the foothold.
Momentum use. Momentum is essential to apply body tension from awkward positions where it’s hard to apply foot force. For example throwing the hips into a plane in which the foot can apply force during execution of a move.
You could go to a gym and train body strength for a decade and it would make little difference to your body tension in climbing if the above factors are not working for you already. The flip side is that many climbers have enough strength already to get a lot more body tension just by working on the technical elements.”