Why do energy levels converge at a continuum and what is a continuum?

In general they don't necessarily converge.

An example is the square potential well: the energy levels do not converge at all and remain perfectly separated.

By "continuum" we simply mean that the energy levels are not distinct and separate (i.e. quantised), but vary continuously as in free space.

However, in the case of the Coulomb potential, for example, the higher energy levels start to get closer together because the potential is becoming wider. And as we approach the maximum value they become infinitesimally closer. Immediately above the largest bound state energy we have the unbound free states, which form a continuum.