Colour developers do two things that are important, they turn exposed silver into elemental silver, and as a side effect they form dyes in the emulsion. These dyes are the reverse of the image colour, in other words blue exposure is a yellow dye, green exposure is a magenta dye, and red exposure is a cyan dye. In a process like C41 or RA4, the bleach turns the elemental silver, back into silver halide, and the fixer then removes all of the silver, but leaves the dye image alone. In RA4 you essentially end up with a negative of a negative which since the colours are reversed, they end up the way the original image was.
E6 works a little differently, in that there are two additional steps. The first developer works similar to a B&W developer, it develops the silver, a chemical reversal step then chemically "exposes" the formerly unexposed silver, which is undeveloped at this point. The next bath is a colour developer which develops the chemically exposed silver and forms the dyes, but leaves the formerly developed silver alone. If you take the film out of the chemistry at this point, it would be black because all of the silver has been exposed and developed. The bleach then turns all of the exposed silver from both steps back into silver halide, so that the fixer can remove it.
In all cases, a stabilizer step is used to set the dyes in the emulsion, this often includes a chemical that makes water slide off the film, similar to what photo-flo does for B&W.