How do we know that Uncle Tom is loved and respected by all the other slaves?

We know that Uncle Tom is looked up to and respected by all the other slaves on Mr. Shelby's estate, first, because the narrator tells us this is the case. When, as readers, we are invited into Uncle Tom's cabin to witness his happy domestic home life, he holds a religious meeting there. Other slaves attend, as does Mr. Shelby's son, George. We are told that:

"Uncle Tom was a sort of patriarch in religious matters, in the neighborhood. Having, naturally, an organization in which the morale was strongly predominant, together with a greater breadth and cultivation of mind than obtained among his companions, he was looked up to with great respect, as a sort of minister among them; and the simple, hearty, sincere style of his exhortations might have edified even better educated persons."

Later, when Tom is going off with the slave-trader to be sold, we learn that,

"A crowd of all the old and young hands on the place stood gathered around it, to bid farewell to their old associate. Tom had been looked up to, both as a head servant and a Christian teacher, by all the place, and there was much honest sympathy and grief about him, particularly among the women."

But actions speak louder than words. What we see of Uncle Tom throughout the novel confirms that he is a person worthy of respect. Beyond his simple, sincere speech, he shows a willingness to sacrifice himself for the other slaves. For example, although he could try to escape, as Eliza does, he chooses not to because he knows this means other slaves--perhaps all the slaves--would have to be sold in his stead to settle Mr. Shelby's debts. He does not want others to suffer.
He is a person of gravitas and moral courage throughout the book, who submits to his body being sold but who dies rather than compromise his soul.