Actually there is no specific time frame was given for example from 1900 to 1999....here only a 5000 years is specified which could be any time period....So in that case Over is preferred over During...Hence E.
Hope this helps.

A-> Grammatically incorrect.

C-> "Beginning a period of" is incorrect.

D-> "During 5000 years" is grammatically incorrect.

Between E and B, B looks grammatically perfect... I will go with "B"

please explain why "during 5000 years" is wrong in this question.
hence why D is wrong.

This is the classic example of meaning and time period.

The sentence is trying to say " the time period of 5000 years after Homo habilis' beginning"

A - Wordy by adding a pronouns "its".

B - changes the meaning. it says tht beginning for 5000 yrs. beginning -- for changes the meaning

C - same as B

D - during - is changing the tense. it makes the meaning continuous, but actually the sentence expresses everything in pat tense.

E - OK

My $ 0.02:

In option B, "beginning" seems to be modifying the actions of Homo Habilis not the period, as such the modifying phrase should be followed by what it is modifying.

"Beginning 2.5 million years ago, Homo habilis roamed the earth, lived in semi-permanent camps, gathered food and shared their economy for a period of almost five thousand years." is a better sentence, but if you move "for a period of almost five thousand years" towards the beginning of the sentence, the continuity is shattered.