You are talking about the Qiraat Styles, the way the Qari recite the Qur'an.
Traditionally, there are 10 recognised schools of qira'at, and each one derives its name from a famous reader of Quran recitation. (Wiki)
There haven't been any musical lessons with notes and rhythms, but they just emerged following the way Prophet Muhammad (peace be upon him) taught his companions.
There is no compulsion though that everyone should recite the Qur'an in one standard style.
Malik Ibn Anas has reported: Abd Al-Rahman Ibn Abd al-Qari narrated: “ Umar Ibn al-Khattab said before me: I heard Hisham Ibn Hakim Ibn Hizam reading Surat Al-Furqan in a different way from the one I used to read it, and the Prophet himself had read out this surah to me. Consequently, as soon as I heard him, I wanted to get hold of him. However, I gave him respite until he had finished the prayer. Then I got hold of his cloak and dragged him to the Prophet. I said to him: “I have heard this person [Hisham Ibn Hakim Ibn Hizam] reading Surah Al Furqan in a different way from the one you had read it out to me.” The Prophet said: “Leave him alone [O ‘Umar].” Then he said to Hisham: “Read [it].” [Umar said:] “He read it out in the same way as he had done before me.” [At this,] the Prophet said: “It was revealed thus.” Then the Prophet asked me to read it out. So I read it out. [At this], he said: “It was revealed thus; this Quran has been revealed in Seven Ahruf. You can read it in any of them you find easy from among them. ( vol. 1 (Egypt: Dar Ahya al-Turath, n.d.), 201, (no. 473))
One of the possible reason is that Qira'at helps stress on words which require listener's attention and to inculcate appropriate emotions pertaining to the verses being recited.
And of course the main obvious reason is that it helps us memorize Qur'an faster that way.
Further knowledge can be gained by listening to 10 different Qira'at style Of Mishari over a verse (Ayat Al-Kursi) from Youtube
May the creator guide us all.