Why did many species of Homo and their ancestors evolve in East Africa?
The question is based on a false opinion but I'll get back to that at the end.

Humans are still evolving


  Why did all species of Homo evolve in Africa first?


Humans are evolving all around the world. Evolution is not a thing that stops once we made the arbitrary decision of naming a lineage. When we first named humans, Homo sapiens sapiens, indeed the lineage was living in east Africa.

Humans is a lineage of Old World Great Ape


  One thing I find interesting about the theory of evolution is the idea that Homo Neanderthalis evolved in Africa


It would help if you could explain why you find that particularly interesting. It is possible that some falsy teleological view of evolution you may have might push you to make some unexpected conclusion from the idea that first Homo lineages lived in Africa (but might be wrong in your understanding of evolutionary processes). In any case, you can find an very introductory course to evolutionary biology here if you're interested.

If humans were a species of feline, it could have evolved elsewhere (it would look very different to what we currently call humans of course) but humans is a lineage of old world great ape. Old world great ape lived mainly in Africa. There is not much more to say about it.


  why Africa did Africa encourage evolution more than other areas of the world?


Africa is a continent. It is big! You can find pretty much any biome in the world in Africa (except toundra and taiga if I am not mistaken). Did the specific location where humans lived in Eastern Africa affect their evolution. Yes, the evolution of a population is always affected by its surrounding. For example, the presence of large game was probably a big selection pressure for cooperative hunting (but you can/could find large game everywhere).

It will be near impossible to answer in the general sense. A lot of things happened in human evolution. Even if you tackle down to a single trait, it will be hard to tell you what would have happened if the world would have been different because it was not. (WorldBuilding.SE welcomes questions like "if it was like that what would have happened?").

Origin of Homo lineages

There are a number of named Homo lineages, some of them have not lived in Africa. This include for example Homo floresiensis.