Negative energy levels in the diagram for a hydrogen atom

We say that a free, unbound electron has zero energy (that's convention, you could just as well put another number there). This means that the level $n = \infty$ is fixed at $E_\infty = 0 \text{eV}$. Since the other levels lie lower, i.e. possess less energy, this forces all other bound states to have negative energies - which then represent that we need to add energy to make the bound state free, which corresponds to raising its energy to zero.

What is the point of having negative numbers, does it somehow aid calculations?

Setting the energy of a free electron to zero does indeed aid calculations because it establishes a convenient reference point. For example suppose you are calculating the energy change in the reaction:

$$ Na + Cl \rightarrow Na^+ + Cl^- $$

Since the energy of a free electron is zero for both atoms the energy change is just the ionisation energy of sodium plus the electron affinity of chlorine.