In each case of a bonded soul, the blood is directly placed against the armor, allowing the iron in the hemoglobin to physically interact with the metal (and presumably any other material it would be bonded with). This means, basically, that the alchemic procedure allows the soul to interact with the metal as well.
When a blood seal is made, it is connected to a being, and must remain with only that being, as well as it must be connected. That means that if Al loses his arm, he cannot move it; however, if he picks up an iron pipe, he cannot move that at free will either, as it was not bonded to him initially. (Even at one point, he transmutes part of his body and can no longer manipulate it.)
(Note: Alphonse has no eyes lit up in his helmet because the helmet is separated.)
In the Fullmetal Alchemist universe, souls are that which contain a being's personality, thoughts, and memories, which is seen later in the series:
For example: Kimblee's soul is able to connect with Pride from within; Hohenheim's many souls are spread around the land and are able to think and act with their energy; and Hohenheim was able to speak and interact with the emotions and arguments of every soul within him from the memories that soul kept.
Thus, we can connect the dots: Alphonse's soul is connected to the armor through the blood seal, and is able to act with his full set of memories; thus, Alphonse can use his memories and thoughts to move the armor at will.
The physical strength and agility comes from their ability to not be restrained by inner forces. (Imagine how much faster you could run if your body worked the same, but without bones and muscles weighing you down!) However, all the fighting skill and ability comes from things that soul has learned (both before getting that body, and after).