Earphone/headphone wires are a PITA because they're so tiny. I'm impressed you managed to separate them and strip the insulation down as well as you did. Normally I just toss them and buy new ones because it's not worth the hassle.
But if you really want to fix it, as described above twist the matching colors together. Get a soldering iron and melt a tiny bit of solder into the twist in each wire to hold it together. (I'd suggest heating the solder directly, instead of the wires like you normally do - with wires this small you could melt the insulation.) Then wrap each individual wire with electrical tape so it doesn't contact the other wires. Then you can wrap the entire 4-wire bundle with electrical tape, overwrapping both ends onto the black cord sheathing for structural strength (when you tug on it, the load should be carried by the exterior black sheath, not the wires inside).