Sefer HaChinuch ספר החינוך

That the king not amass many horses: That the king that is ruling over us not amass many horses for himself, besides those needed for his chariot and the chariots of his horsemen; and as it is written (Deuteronomy 17:16), “he shall not amass many horses for himself.” And the content of the commandment is that he should not have horses that run in front of him simply for his glory. And even over one idle horse does he transgress this. However if he has horses in his stables prepared for war and sometimes his horseman ride upon them, this is not at all included in this negative commandment; as the central matter is only from the angle that we said — that he should not have idle horses always go before him for stature and glory, but rather he should have only one animal upon which to ride.

And the verse already stated the reason of this commandment, which is that he not “bring back the people to Egypt in order to amass many horses, etc.” — meaning to say, that he not send from his people that they should establish their residence in Egypt to raise horses there. As the prohibition is only to establish residency there, as they, may their memory be blessed, expounded (Talmud Yerushalmi Sanhedrin 10:8), “To dwell you shall not return, but you can return for business.” And [it is] also so that his heart not become proud by amassing many horses, silver and gold; since the horses’ masters become proud due to their beauty and speed.

The laws of the commandment are in the second chapter of Sanhedrin.

And this prohibition is practiced at the time that Israel is in its inhabitation and that we have a king. And this is from the commandments that is only upon the king. And a king that transgresses it and adds even one free horse to be running in front of him — in the way that the other kings of the nations of the world do — is liable for lashes.