Ok, the Rydberg formula was originally an empirical formula determined for the spectral lines of a hydrogen atom. As you probably know, the hydrogen atom can only absorb certain frequencies of light which more or less match the allowed energies levels of the electron as it orbits the positively charged nucleus. There are an infinite number of allowed states but they get closer and closer together as the energy increases, eventually converging to infinitely small separations as the ionization limit is reached - that is, when the electron is at infinite separation from the nucleus. The Rydberg formula can be extended to any "hydrogenic system", i.e., one in which a positive nucleus is surrounded by a single electron. E.g., He+ is a hydrogenic system, with a different core charge.
In any atomic or molecular system, an electron experiences similar quantization of its allowed energy levels. These energy levels are usually quite a bit more complicated, and in most cases can't be determined analytically, because there are a lot more interactions to worry about. BUT, when an electron in any system is sufficiently excited, every system can be approximately reduced to a hydrogenic system, because the excited electron is far away in terms of both energy and physical separation from the rest of the molecule that it basically becomes a positively charged center with one electron orbiting it. The electronic states at that point can therefore be approximately described by the Rydberg formula, so we call these states "Rydberg states". This is frequently only the case near the ionization limit.
For hydrogenic systems, the energy of a quantized level in an idealized neutral space depends only on the principle quantum number n. This is implicit in the Rydberg formula. Which is to say, the energy does NOT depend on the angular momentum of the electron, even though any given shell may comprise several states of the same principle quantum number (e.g., n = 1) but different orbital and spin angular momentum values (l = -1, 0, 1). This breaks down in the presence of external electric and magnetic fields. Because electrons are charged, if they have nonzero angular momentum then they have energetic interactions with external electromagnetic fields. In practice this causes a splitting of the normally degenerate (equal energy) states for each principle quantum level. For instance, while all three states (orbitals) that make up the n = 1 level of hydrogen have identical energy, when placed in an electric or magnetic field, they split, with one of the states being increased in energy, one being decreased in energy (depending on their spatial interaction with the field and the field's strength), and one (the one with no angular momentum) staying the same. This is called either the Stark effect or the Zeeman effect, depending on whether the field is electric (Stark) or magnetic (Zeeman). The Zeeman effect is probably better known because it is essentially responsible for NMR and MRI.
Since Rydberg states are composed of multiple states that have different allowed values of orbital angular momentum, they would split in the presence of an electric field due to the Stark Effect. Spectroscopically, it results in a peak splitting (triplets, doublets, etc.) in most ordinary situations. My understanding of ZEKE is that when Ryberg-states that are near the ionization threshold are subjected to electric fields, Stark effect splitting causes the ionization threshold to be reached, causing ionization. By modulating the electric field after excitation of molecules to Rydberg states, ionization energies can be therefore determined with high precision and accuracy compared to conventional photoionization techniques. But this type of spectroscopy is not my specialty so that's probably a fairly superficial explanation.
Does this help?