The basic gist is that the knowledge from the ancient world was primarily spread through the Roman Empire, who were heavily influenced by the Greeks, who took and built upon the Egpytians and other mediterranean/middle eastern civilisations. The western european part of the Roman Empire collapsed for various reasons, and europe regressed in the middle ages (also called the dark ages). However the eastern part, which became the byzantine empire stuck around for a long time and did a lot of dealings with the arabs, and the beginning of islamic nations. Most of the knowledge was centered in Constantinople.

Arabs and Islam spread through northern and eastern africa, and to a lesser degree some parts of the west, where they dealt and integrated with native sub-saharan africans, such as the Mande. This resulted in states like Mali, where the famous Timbuktu university existed. Islam brought with it the knowledge from the GrecoRoman empires, and established trade routes. The caliphates and other islamic states in africa were generally more advanced than medieval europe.

Along with a couple of instances of some texts making it to europe via the crusades, the ottoman empire (an islamic caliphate) grew in power enough to seize Constantinople, which caused most of the scholars to flee back to europe. This combined with the reconquista in spain, which resulted in more arabic greek-inspired sciences being deciphered in europe. European cultures began to integrate this knowledge into their societies, and was a huge factor in the Renaissance, the Age of Enlightenment and the split of Christianity with Protestants from Catholics, which was a factor in a lot of smaller scale conflicts (which also drove forward science and technology). The first wave of colonialism occured, which was mostly focused on the Americas, and the Asian subcontinent. A little bit of South East Asia too but I might be wrong on that.

So all this new knowledge was advancing europe rapidly, and they were expanding trade and resources but Islamic states basically stagnated because they saw themselves as superior. This was expounded upon because of the dominance of Sunni Islam - which was pretty much unchallenged - and resulted in a much more intellectually conservative society. Islamic societies started to regress like europe had before, whilst the world around them was changing fast, important trade routes and harbors became less relevant as europe colonised. Their power waned and they basically couldn't do anything when europe had advanced past them, and some last ditch efforts to modernise caused even more resentment among the populace towards european/western culture.

Europe went and colonised pretty much all of Africa whilst the Ottomans tried not to fall apart and shrunk continously, and most of the Islamic states (like Somalia) that were allied with the Ottomans weren't much better off. A lot of central and southern Africa was mostly untouched by most of the rapid development over the last few centuries too, so they couldn't do much against the colonists either. Your knowledge of history probably kicks in fine here, but basically after the first world war, the ottoman empire totally collapsed, and europe started splitting up the middle east and africa, resulting in the mess we've got today.

I'm wayyyy less knowledgable about east asian (chinese, japanese, etc) history but they mostly developed independantly as far as I'm aware, so they don't really have the same roots as we do.