My answer is E. I'll try to explain this with my limited knowledge of grammar.

First let's talk about "that" versus "of". "That" is used to connect two sentences. "of" is a preposition which is used to connect two nouns, pronouns or phrases. In this question we have two sentences:

A "calendar stick" carved centuries ago by the Winneba tribe may provide the first evidence

and

the North American Indians have developed advanced full-year calendars basing them on systematic astronomical observation.

Using "of" in place of that, Option B and D are ruled out (coz. they refer to "evidence not of development but of North American Indians". That's weird.)

Option C would have been correct had the author not used "basing them". Please note that the use of "of" is not wrong. In all the options (B, C and D), "of" is used to compare two phrases - 'the first evidence' with 'the North Americans' or 'the development'.

Hope this helps.

- 1st 2-3 split: 'basing them on' vs 'based on' . 'based on' is correct idiom -> A,B, C are out.
- 'evidence of' vs 'evidence that' . Both terms are correct. And D and E expresses the same meaning but E is clearly concise.
Therefore E is answer