An electron in an atom is like a note on a string: it has a set of allowed energies, and the physicist's task is to find these notes. Usually, the space around the nucleus is divided into small pieces and the behavior of the electron in each piece is calculated. The finer the pieces, the more accurate the answer. But for the hydrogen atom, accuracy hit a ceiling: no matter how fine the division, the error would not drop below a certain level.
The reason turned out to be in how the program looks at the center of the atom. At the very nucleus, the electron feels a force that grows without limit, and the standard calculation did not account for this. The authors added knowledge about what happens near the nucleus to the program, and accuracy immediately improved by tens of times. Now the energies of electrons in the hydrogen atom can be calculated with an error of less than one billionth.
The same trick works for other atoms where electrons are strongly attracted to the nucleus. Calculations become faster and more accurate, which is important for understanding chemical reactions and material properties.
The error in calculating the electron energy in the hydrogen atom was like measuring a person's height to the nearest millimeter but failing to notice they were standing on a stool.