Hirai Sensei, the founder of Korindo Aikido once said:
"when training empty handed, imagine you are training with a weapon. When training with a weapon, imagine you are empty handed."
Empty hand and weapons practice are closely related. The longer one trains with weapons, the better one can realize this. each weapon one practices, enhances other specific features that are related to Aikido. The Ken (sword) is excellent for learning to move with whole your body, from the center outwards, as well as deeper understanding of Mai (distance) and center line, the Jo is excellent for side to side movement and flowing, and enhances the further the Mai (distance) by having a different one and the concept of center line and line of attack. The Bo (RokShakuBo) enhances the hip movement - having more weight to lift and throw around. The Nito (two swords practice) teaches better control of the body and spliting ones force to two separate simultaneous actions and recombining (on the latter, I can only speculate, I hardly practiced it).
As for why one trains with weapons. The answer is the same for all other elements of the art - it is a part of the M.A. you love and it is great fun