Let's take the simplest case, that of a hydrogen atom. When the electron of a hydrogen atom is given a blast of energy, it rises to a higher energy level (i.e. it moves further away from the nucleus). When the excited electron then loses this energy, it relaxes back to a lower energy level. The energy released as the electron relaxes back to the lower energy level is what produces the line emission spectrum.
Now, why is it a series of lines, rather than just one line? After all, there is only one electron in a hydrogen atom, so why not just one line? The answer is that there are several ENERGY LEVELS within the atom. So, when the electron is excited by the extra energy, it can climb to any one of several energy levels. Thus, when it relaxes back to a lower energy level, it has a number of levels where it can end up, giving out differing amounts of energy depending on which level it relaxes to. The lines you see in the spectrum correspond to these differing amounts of energy. The reason it is a series of lines and not a continuous band is because these energy levels only exist at certain values (or certain distances away from the nucleus). Therefore, only certain packets, or "quanta" of energy can be emitted. Finally, the lines in the spectrum converge as you move towards the blue end of the emission spectrum (high frequency). This is because the gaps between the separate energy levels get smaller and smaller as the electron moves further away from the nucleus. You might wish to think of it as a series of rungs on a strange ladder, in which the rungs get closer together as you go up the ladder.