There are different opinions regarding the actual meaning of the animal sacrifices. Maimonides (1135-1204) understood the sacrifices as a divine concession. G-d recognized that when the Israelites left Egypt they were steeped in a religious culture that brought sacrificial offerings to their gods. G-d decided, so to speak, to allow the Israelites to express their religious passion in a somewhat similar way that they were accustomed to with some limitations which were meant to help wean them off this approach towards the eventually annulment of animal sacrifices.
Thus Maimonides writes:
Many precepts in our Law are the result of a similar course adopted by the Supreme Being [i.e. gradual evolution]. It is impossible to go suddenly from one extreme to the other; it is therefore, according to the nature of man, impossible for him suddenly to discontinue everything to which has been accustomed . . . The custom which was in those days widespread among all people, and the general mode of worship in which Israelites were brought up, consisted in sacrificing animals in temples which contained certain images, to bow down to those images and to burn incense before them . . . For this reason, G-d allowed these kinds of service to continue. He transferred to His service that which had formerly served as a worship of created beings [i.e. idolatry] . . . By this Divine plan it was effected that the traces of idolatry were blotted out, and the truly great principle of our faith — the existence and unity of G-d — was firmly established. This was achieved without deterring or confusing the minds of the people by the abolition of the service to which they were accustomed and which alone was familiar to them.
In other words, although G-d was not at all interested in sacrifices G-d conceded to the needs of the people understanding where they were at. However, this concession is nonetheless with an eye towards the future when the ideal Temple worship will be prayer.
Over time the Temple services will evolve and ultimately animal sacrifices will be nullified. Rabbi Abraham Isaac Kook (1865-1935) elaborates that when the Temple is reinstated, there will be a period of time when we will no longer be slaughtering animals for consumption or pleasure, and eventually we will only bring meal offerings. But even the meal offering will be nullified until finally the Temple will be strictly a house of prayer.
In other words, when we reach the point when we no longer need the sacrificial service, we will be on a high enough spiritual level to experience “giving ourselves completely to G-d” through prayer alone. We will put our entire souls into our words of prayer – recite them soulfully — and thus no longer need sacrifices to experience our connection G-d.