Let's look at this in its simplest form, the case of the hydronium and ammonium ions. Water and ammonia are neutral molecules (each have 10 protons and 10 electrons). In water, four of these electrons are in lone pairs, or sp3 orbitals on the oxygen. In ammonia, two of the electrons are in a lone pair. If we add Brønsted Lowry acid (which is in essence free or easily freed protons), a proton will add to the negatively charged lone pairs, form a bond (mixing its empty 1s orbital with the filled sp3 orbital to form a lower energy σ orbital). No electrons have been added, and a proton has, so this new species (H30+ or NH4+) has a positive charge.
Another way to think about this is that when the nitrogen or oxygen uses one of its lone pairs to bond with hydrogen or something else (in the case of the nucleophilic substitution, carbon), one of its electrons 'goes over' to the other atom, changing its charge by +1. Now, this isn't exactly what happens, but it is a useful approximation in most cases and is what we use to determine formal charge, which is defined as the difference between the number of electrons on an atom and the number of protons it has.