Their outer electron shells (valence electrons) are full.
Valence electrons are the electrons in the outermost shell, an atoms are stable and "happy" when they have full outer shells. The first three shells have 2, then 8, then another 8 electrons. Noble gases have full outer shells already (they are in group 18, which means that they have 8 valence electrons, a full shell) , so they have no need to react in order to fill them up, while other elements want to take some or give a couple away. Helium has only 2 valence (and total) electrons, but because they fill up the innermost energy level, it is also inert.