Caleb's quieting of the the people in defense of Moses is the most likely reason for God speaking of Caleb so approvingly. The Talmud provides some details that shed light on the the opinion of later rabbis such as Ibn Ezra and Rashi cited by @user22655. In this passage the people silence Joshua, but Caleb succeeds in speaking out and convinces them to support Moses. From Sotah 35a:

The verse states: “And Caleb stilled the people toward Moses” (Numbers 13:30). Rabba says: This means that he persuaded them with his words... How did he do so? Joshua began to address the people, and as he was speaking they said to him: 'Should this person, who has a severed head, as he has no children, speak to the people about entering Eretz Yisrael?'

Caleb said to himself: 'If I speak they will also say something about me and stop me from speaking.' He began to speak and said to them: 'And is this the only thing that the son of Amram, Moses, has done to us?' They thought that he wanted to relate something to the discredit of Moses, and they were silent. He then said to them: He took us out of Egypt, and split the sea for us, and fed us the manna. If he says to us: Build ladders and climb to the heavens, should we not listen to him? “We should go up at once,” even to the heavens, “and possess it” (Numbers 13:30).

The opinion or Rashi is based on Sotah 34b

It is also stated with regard to the spies: “And they went up into the south, and he came to Hebron” (Numbers 13:22). Why is the phrase “and he came” written in the singular form? The verse should have said: And they came. Rava says: This teaches that Caleb separated himself from the counsel of the other spies and went and prostrated himself on the graves of the forefathers in Hebron. He said to them: My forefathers, pray for mercy for me so that I will be saved from the counsel of the spies.

Interestingly Rava also gives the opinion that God praised Caleb because he was willing to change his mind. Here "having another spirit with him" is interpreted as having once sided with the faithless spies.

As it is stated: “And Moses called Hoshea son of Nun Joshua” (Numbers 13:16), meaning: 'God will save you' from the counsel of the spies. And this is the meaning of that which is written: “But My servant Caleb, because he had another spirit with him, and has followed Me fully, him will I bring into the land where into he went” (Numbers 14:24), which implies that Caleb changed his mind over time. Joshua, however, was opposed to the intentions of the other spies from the outset.

Summary: the rabbis of the Talmud explain God's praise of Caleb as the result of

• Caleb's quieting the people after Joshua had been effectively silenced

• His prayer at Hebron at the graves of the forefathers

• His willingness to change his mind after at first having the wrong opinion

Since the first of these is supported by the clearest biblical evidence [Numbers 13:30], this explanation deserves the most weight.

Caleb's quieting of the the people in defense of Moses is the most likely reason for God speaking of Caleb so approvingly. The Talmud provides some details that shed light on the the opinion of later rabbis such as Ibn Ezra and Rashi cited by @user22655. In this passage the people silence Joshua, but Caleb succeeds in speaking out and convinces them to support Moses. From Sotah 35a:

The verse states: “And Caleb stilled the people toward Moses” (Numbers 13:30). Rabba says: This means that he persuaded them with his words... How did he do so? Joshua began to address the people, and as he was speaking they said to him: 'Should this person, who has a severed head, as he has no children, speak to the people about entering Eretz Yisrael?'

Caleb said to himself: 'If I speak they will also say something about me and stop me from speaking.' He began to speak and said to them: 'And is this the only thing that the son of Amram, Moses, has done to us?' They thought that he wanted to relate something to the discredit of Moses, and they were silent. He then said to them: 'He took us out of Egypt, and split the sea for us, and fed us the manna. If he says to us: Build ladders and climb to the heavens, should we not listen to him? We should go up at once, even to the heavens, and possess it.' (Numbers 13:30)

The opinion of Rashi is based on Sotah 34b

It is also stated with regard to the spies: “And they went up into the south, and he came to Hebron” (Numbers 13:22). Why is the phrase “and he came” written in the singular form? The verse should have said: And they came. Rava says: This teaches that Caleb separated himself from the counsel of the other spies and went and prostrated himself on the graves of the forefathers in Hebron. He said to them: My forefathers, pray for mercy for me so that I will be saved from the counsel of the spies.

Interestingly Rava also gives the opinion that God praised Caleb because he was willing to change his mind. Here "having another spirit with him" is interpreted as having once sided with the faithless spies.

As it is stated: “And Moses called Hoshea son of Nun Joshua” (Numbers 13:16), meaning: 'God will save you' from the counsel of the spies. And this is the meaning of that which is written: “But My servant Caleb, because he had another spirit with him, and has followed Me fully, him will I bring into the land where into he went” (Numbers 14:24), which implies that Caleb changed his mind over time. Joshua, however, was opposed to the intentions of the other spies from the outset.

Summary: the rabbis of the Talmud explain God's praise of Caleb as the result of

• Caleb's quieting the people after Joshua had been effectively silenced

• His prayer at Hebron at the graves of the forefathers

• His willingness to change his mind after at first having the wrong opinion

Since the first of these is supported by the clearest biblical evidence [Numbers 13:30], this explanation deserves the most weight.