Explain how is it possible that each species has the reproductive potential to spread over the earth, and yet populations remain more or less stable for most species?

When it comes to population growth, reproductive potential is not the only factor. In absence of any hindrance, species would grow indefinitely and show exponential growth (J shaped). However, in real life species show logistic growth (S shaped). Many factors contribute towards this restricted and stable population for most species.

Every species has a carrying capacity in a particular environment. This means that there is a fixed number of organisms from that species that can be supported by the given environment. After this number, the population becomes more or less constant because of limited resources like food and shelter. Intra-species Competition also eliminates less fit individuals further bringing down the population size. Predation by other species and inter species competition for resources also keeps the population in check. Birth rate and death rate for a given species keeps it stable.