The Hydrogen atom is fairly well explored, and has a nucleic charge of +e, while the electron orbiting it has a charge of -e. As the electron becomes excited, it moves farther away from the nucleus until it is finally ionized.
In more complex atom, such as Rubidium, you have a nucleic charge of +37e, and 37 electrons with a charge of -e surrounding it. When observed from a distance, these add to a net charge of 0, making it neutral.
If you are able to excite the outermost electron of an atom, the electron will move very far away from the nucleus, such that the nucleus and the other electrons are close together compared to the distance between the excited electron and the nucleus. From this outer electron's perspective, the other parts of the atom combine to form a net charge of +e. With a charge of -e orbiting a charge of +e, this atom will behave like a Hydrogen atom. This is a Rydberg atom.
What they have done is make a molecule out of two Rydberg atoms made from Rubidium. This should be similar to two Hydrogen atoms making a H2 molecule. Very excited electron energies are unstable and have very brief lifetimes, so it's unusual that the entire molecule would have lasted for 18us.