Why does Locality Sensitive Hashing use multiple sets of hash tables? How does it guarantee similarity?
LSH indexing first concatenates $k$ simple hash functions $h$ to produce a function $g$. Note that $g$ is a regular LSH function, only with slightly different parameters. In particular, probabilities of collision decrease for both near and far objects. The new probabilities are $p'=p^k$.

Then they take $L$ such functions $g$ and produce $L$ hash tables. Then we search through all the hash tables, one by one. This effectively increases the probability of collision. The new probabilities are $p''=1-(1-p^k)^L$. Note that this function has a sigmoid shape.

It may seem weird to first reduce $p$ and then increase it. However, the parameters $k, L$ are useful knobs for controlling the behavior of the algorithm. You could certainly run with $k=L=1$, but larger values may perform better.