This question has been asked a couple of times before, but the answers are from years ago, so here's a slightly fuller explanation, though there's really not much to say.

Although the scene in Kingdom of Heaven is not a historical event, it is based on one. According to contemporary sources, Saladin sent Richard the Lionheart a box full of ice toward the end of the Third Crusade. He did this as a diplomatic gesture when he heard Richard was unwell and thought the ice might help with the king's fever. He also sent bowls of exotic fruit because... Saladin just liked bowls of fruit. According to Ibn-Shaddad, one of Saladin's biographers, Richard sent more requests for ice and fruit, especially pears and plums. The ice came all the way from Mount Hermon, part of the mountain range that covers much of the modern border between Lebanon and Syria. It was one of very few mountains in the region that regularly got snow and ice, and blocks of it were brought down from the mountain for people to use. There's no complicated explanation for how they got the ice - they literally walked up a mountain and fashioned blocks of it from what they could find. Ice wasn't made, it was mined and quarried.

However, how they kept the ice cool for the long journey from Mount Hermon to Richard's camp all the way by Jaffa, which would have been several days of continuous travel, we really aren't sure. We know from ancient sources how ice could be stored - in antiquity ice was stored in underground caverns lined with insulating material, essentially turning the cavern into a giant cooler - and they were probably still doing that in the Middle Ages. It's likely that the block of ice was cut to be large enough that, even after several days, it was still a significant block of ice. They could have done things like put it in a box packed with snow and insulation, or travelled by night and burying the ice in small dug out caverns during the day to protect it from the heat, but we don't know.

This was not meant to be a sustainable trade in ice, however. Ice was not regularly transported these kinds of distances. It was a one off gesture to paint Saladin in a good light while also showing Richard I how powerful he was. Heatstroke was a serious problem for crusading armies, and here was Saladin gifting people ice. They didn't need to know that it was probably an enormous hassle to do, just that Saladin had the power to dispense ice in a desert. That Richard apparently sent requests for more shows that it worked.

u/J-Force's answer and the older answer from u/Valkine are good explanations about how ice was transported from the mountains, but I just wanted to point out that the scene in Kingdom of Heaven is actually a real event, which is described in detail independently in Christian and Muslim sources, and this scene in the movie follows them pretty closely. And I wrote something about it recently, if you're interested!