Augmented sixth chords can be scary because they are typically the first non-triadic chord a student runs into. To understand how they work, you have to set aside your understanding of triads and think about the chord from a purely voice leading perspective.
We know from triadic harmony that half steps can create a sense of expectation when they resolve to their proper note. The augmented sixth utilizes this principal to an extreme degree. In the key of C, we want a chord that very strongly leads into our Dominant, the G chord (in other words, we want a really powerful Predominant chord). Thinking purely from a voice leading perspective, the most powerful thing we could do harmonically to lead into G would be to have both F#, a half step, and Ab, another half step, both resolve up and down respectively. That is the basis for our augmented sixth chord.
So we have our Ab (traditionally as the bass note) and our F#, but we need some more notes to fill out that sound! Well, composers tried a few. Adding the tonic makes sense, since it's a pretty important note, so we'll call that the Italian augmented sixth. When we add the second above the tonic, we'll call it French, and when we add the minor 3rd, we'll call it the German augmented sixth. Don't take those names too literally, as tradition is really the only reason why we still use them.