Why is it wrong to eat meat from a living animal? Is it because it is cruel? Is it because it is impure? Why is it?
Rambam explains this prohibition as follows:
Guide for the Perplexed 3:48
It is prohibited to cut off a limb of a living animal and eat it, because such act would produce cruelty, and develop it: besides, the heathen kings used to do it: it was also a kind of idolatrous worship to cut off a certain limb of a living animal and to eat it. (Friedlander translation)
The Sefer Hachinuch as well links it to cruelty, going so far as to say that there is no greater cruelty in the world:
Sefer Hachinuch Mitzvah # 452
משרשי המצוה כדי שלא נלמד נפשנו במדת האכזריות שהיא מדה מגונה ביותר ובאמת שאין אכזריות בעולם גדול ממי שיחתוך אבר או בשר מבעל חיים בעודנו חי לפניו ויאכלנו
R Jack Abramowitz answers in the following way
The reason for this is similar to that for the prohibition on eating blood: when a limb is severed from a living animal, it still contains the animal’s life force. This is also evident in the verse stating the prohibition to Noah: Flesh with its life…you shall not eat (Genesis 9:4). To eat a limb torn from a live animal is unspeakably cruel. Since we are what we eat, eating such a thing only reinforces the trait of cruelty.