Rabbi Aqiva and Ben Azzai have a dispute as to what is the overriding principle behind the Torah. According to R' Aqiva, it is "love your peer as yourself", and according to Ben Azzai, it is "these are the generations from Adam" -- ie the common brotherhood of all people. (Yerushalmi, Nedarim 9:4)
So your question may be very much like "But aside from that, Mrs Lincoln, how did you like the play?"
Why love other people? Because the definition of a human being fulfilling their purpose is that they love other people. The answers to "why avoid damaging others?", "why be respectful?", or "why pursue self refinement / sanctity?" have to be framed in terms of how they help us love other people.
More recently, in the early 19th cent, R' Chaim Volozhiner wrote the notebooks for his son R' Yitzchaq to compile into what became "Nefesh haChaim". A Qabbalistic work about how people can impact the metaphysics of the universe(s) and thus change the events of history (among other things), and how action, prayer and Torah study can repair the world. Usually read as most centrally focusing on the importance of Torah study. But, R' Yitzchaq Volozhiner writes in his introduction to the work:
He would routinely rebuke me because he was that I do not share in the pain of others. This is what he would constantly tell me: that the entire person was not created for himself, but to be of assistance to others, whatever he finds to be in his ability to do.
Skipping ahead another century, we get to the introduction to Shaarei Yosher, written by R' Shimon Shkop. (The translations in this answer come from a more recent manuscript than the translations in the PDF.) The introduction opens:
יתברך הבורא ויתעלה היוצר
שבראנו בצלמו ובדמות תבניתו,
וחיי עולם נטע בתוכנו
שיהיה אדיר חפצנו,
להיטיב עם זולתנו,
בדמות הבורא כביכול
Blessed shall be the Creator,
and exalted shall be the Maker,
Who created us in His “Image” and in the likeness of His “Structure”,
and planted eternal life within us,
so that our greatest desire should be
to benefit others,
to individuals and to the masses,
now and in the future
in imitation of the Creator (as it were).
He explains how the very definition of qedushah (holiness) is one's commitment to that goal: "That is, that we, the select of what He made –
should constantly hold as our purpose to sanctify our physical and spiritual abilities,
for the good of the many,
according to our abilities.
In my opinion, this whole concept is included in Hashem’s mitzvah 'Be holy, [for I am Holy].'" Be holy, ie be committed, in imitation of G-d's commitment.
But people were created with a healthy self-interest, without which we wouldn't be motivated to value our own actions and what we make. Without self-interest, we would lack the motive to contribute to the good of others.
So how does Rav Shimon suggest this dialectic can be resolved? By realizing that giving is not based on abnegating the self, but by extending it to include others:
The entire “I” of a coarse and lowly person is restricted only to his substance and body. Above him is someone who feels that his “I” is a synthesis of body and soul. And above him is someone who can include in his “I” all of his household and family. Someone who walks according to the way of the Torah, his “I” includes the whole Jewish people, since in truth every Jewish person is only like a limb of the body of the nation of Israel. In this [progression] there are more levels for a person who is whole, who can connect his soul to feel that all of the worlds are his “I”, and he himself is only one small limb in all of creation. Then, his self-love helps him love all of the Jewish people and [even] all of creation.
So, we are good to others by removing that otherness. We are all parts of a single whole. A unity that is the "shadow" of G-d's, a love and lovingkidness that emulates His.
If the Ramchal tells us we are here in this world to perfect ourselves, Hillel, R' Aqiva, Ben Azzai et al tell us that what function a person has that we are to be perfect at -- being good to others.
Being connected to others is imitating the Divine, it is holiness, it is doing what we're made for. It's the ultimate "because" behind every why.