In Islam lying on the belly or lying with your belly down is forbidden. Why is that so? Why sleeping with the belly down is forbidden? What is the narration forbidding that?
First I couldn't find such a general statement saying that lying on your belly or lying with your belly down is prohibitted. But the literal wording of the ahadith is about sleeping in such a position!
It is rather makrooh (frowned upon) to seek this position or that of lying on your left side when starting to sleep. While it's certainly not sinful to do change a position while sleeping as we know that what one does while sleeping is not written in your record of deeds:
There are three (persons) whose actions are not recorded: a sleeper till he awakes, an idiot till he is restored to reason, and a boy till he reaches puberty. (Sunan abi Dawod and in similar narrations all Sunan books)
Secondly as for the reason why this is considered as bad you may refer to the hadith quoted by imam Ahmad which was quoted in the fatwa provided by @servant-of-wiser on islamqa or the long version of the hadith in Sunan abi Dawod or the version imam al-Bukhari compiled in his al-Adab al Mufrad with the following wording:
Ibn Tikhfa al-Ghifari reported that his father told him that he had been one of the People of the Bench. He told him, "I was sleeping in the mosque during the last part of the night, lying on my stomach. Someone came to me and moved me with his foot, saying, 'Get up. This is a manner of lying down which Allah hates.' I raised my head and the Prophet, may Allah bless him and grant him peace, was standing by my head."
Aren't we asked to leave anything which Allah hates? In another version of the hadith it was mentioned as the "manner of lying down of the people of hellfire", this hadith was mentioned by ibn Majah in his Sunan and could be a reason to genetalize the ruling:
"The Prophet(ﷺ) passed by me and I was lying on my stomach. He nudged me with his foot and said: 'O Junaidib! This is how the people of Hell lie.'""
There are other ahadith on this topic too in Jam'i at-Tirmidhi and Sunan ibn Majah (with the wording "face down").
Note that all of the ahadith on the topic have been discussed and checked by al-Hafidh Ibraheem ibn Ishaaq al-Harbi إبراهيم بن إسحاق الحربي (a teacher of imam Ahmad) in his book called Ikraam a-Dayf إكرام الضيف ("welcoming of the visitor") and he pointed at several weaknessses in them. Imam al-Bukhari and ibn abi Hatim also considered them to be weak and most of the weak narrators Ibraheem ibn Ishaaq al-Harbi pointed at are considered as weak by these two scholars. So there's a certain difference on opinion on these ahadith and the ruling. Sheikh al-Albani (note that he quoted the hadith from Sunan ibn Majah in his da'if at-Targheeb wa at-Tarheeb) and Muhammad Salih al-Munajjid for example are among those modern scholars who considered it as sahih.
So in general some scholars consider the whole lot of ahadith on lying the belly down as da'if in general while others consider it hassan by the sum of narrations on the topic.
Scholars concluded from the above narration that this manner of lying down is at least frowned upon I found a fatwa (in Arabic) of Sheikh ibn Bazz saying so (but I found statements saying that he doesn't consider the ahadith as sahih), and here on islamweb another one (in Arabic) with comments of ibn Qayyim al-Jawziyyah and Dar al-Ifta' in Jordan provides statments from Shafi'i fiqh saying so.