You can think of it in terms of attraction - positive and negative charges are attracted to each other.
In a neutral atom, the number of electrons is equal to the number of protons in the nucleus, creating a net charge of zero for that atom.
Remove one electron from the atom, and now you have the original number of protons attracting one fewer electrons, and the net charge of the atom has gone from 0 (neutral) to +1 (positive).
The attraction to the nucleus felt by each remaining electron is stronger in the postive atom than in the neutral one because you have the same strength of positive charge being distributed over fewer electrons. The more electrons you remove from an atom, the more energy will be required to remove the next one - all because the pull of the positive nucleus remains unchanged, and is distributed across fewer and fewer electrons.