Understanding the q-SDH Problem (on elliptic curves)
The goal of the problem as you've described it is to find any $x$ along with $\frac{1}{\beta + x} g_1$.

There's no real reason it's in the denominator. That's just the definition of the problem. If you believe the assumption that this problem is hard, then you believe that finding those values (where $x, \beta$ are in the denominator) is hard to compute.

As for why it's hard: we don't know that it's hard, it's just an assumption (as one comment pointed out, it's definitely not harder than the discrete log problem).