The Ben Ish Chai in his sefer אמרי בינה here says that the reason why Adam and Noach and the Avos brought animal sacrifices was to bring part of themselves as a gift to Hashem through the thoughts that they had when they brought the offering. That is, when they were slaughtering the animal on the altar they imagined that they themselves were being slaughtered and offered up to Hashem, and the power of these thoughts rose up to Hashem and dragged along with them the holy sparks that are intermixed in all parts of Hashem's creation - humans, animals, vegetation and minerals.
This is the gift which a person brings to Hashem with the sacrifices, because imagination alone is not enough to separate the holy sparks - both a physical action together with the thoughts of the imagination are needed, as we see by many mitzvos, for example, tashlich, shofar and lulav. Also, we see many times in the Tanach that the prophets acted out various physical procedures in order to effect certain prophecies.
This elevation of the holy sparks through the thoughts of the imagination that one has when bringing a sacrifice is similar to the elevation of the holy sparks that happen when we imagine giving our lives for the sake of the sanctification of Hashem's name when we recite the word אחד at the end of the first posuk of the Shema, and similarly when we recite Tachanun.