The terms simple and compound meter are really misnomers, as Paul Creston explains in his Principles of Rhythm--and there are other myths to dispel.
First, a time signature is not an instruction; it isn't telling you to do anything. Measures measure off sections of the music (hence the name) according to how much time they occupy. Time signatures tell you how much time that is. A piece of music will sound the same no matter where it puts its measure lines and what time signature it uses. (Don't accent the first note following a measure line unless there is a good musical reason to do so.) It makes sense, however, to measure off sections of music according to groupings that we can actually hear, so long as we're not pedantic about it (changing time signatures every time the grouping changes no matter how briefly, for example).
For this reason, a time signature of 2/4, for example, means there are two quarter notes worth of time in each measure and also implies that we'll hear groupings of two. Two what? Well, two beats or pulses we usually say. It also implies that each beat or pulse will itself divide into two equal parts. Why? Because that's the default.
Suppose we want each of the two beats to divide into three equal parts. That's easy enough to play, but not so easy to notate. If we represent the beats as quarter notes, what do we represent the parts as? Twelfth notes (1/4 divided by 3 = 1/12)? Unfortunately (triplets aside), we have no twelfth notes.
Suppose instead we represent each of the two beats as dotted quarters. Now we can represent the parts as eighth notes. Fine, but what will be our time signature? Since we don't have a numeral with which to represent a dotted quarter note, we'll have to count up how many parts we have altogether. Each of the two beats has three parts, and that gives us six parts altogether. The parts are represented as eighth notes, and thus our time signature is 6/8, which implies a grouping of two beats, each of which is in turn divided into three parts.