Why $o(\log n)$ bits instead of $O(\log n)$ bits needed for succinct data structure?
Note that $\lceil \log(2^n) \rceil = n$.

An algorithm that uses $n$ bits to represent the set would be fine (it would count as a succinct data structure).  An algorithm that uses $n + 2 \sqrt{n}$ bits would also be fine.

An algorithm that uses $2n$ bits to represent the set would not be fine (it would not count as a succinct data structure).  Similarly, an algorithm that uses $1.01n$ bits would not be fine.

This is just what the definition says.  You could imagine some other definition, but that's what the standard definition of "succinct data structure" is saying.  It is saying that you can use a little bit more space than required by the information-theoretic lower bound, but not too much -- not a multiplicative factor.