The random densification technique-JL lemma In Ailon's paper (p.3): How $1/(20nd)$ is obtained?
It's not "obtained", but rather the bound the authors want on $\mathrm{Prob}[|u_1|\ge s]$. The Chernoff inequality says how large $s$ needs to be in order to guarantee the desired upper bound. As they assume $d \le n$, it suffices for $s$ to satisfy $s^2 \cdot d/2≥\ln(20n^2)$, which leads to $s=c\cdot d^{-1/2} \sqrt{\log n}$ for some appropriately chosen constant c.