The wires bulged out of my headphone cord. How can I fix this, without solder?
Since that appears to be a coaxial headphone cable, it's unlikely there's a good way to fix the damage shown. If you have special tools, it might be possible to cut completely through the cable, strip appropriately, and crimp the conductors back together, but it isn't practical to buy the tools for a single cable, or for a consumer (and you'll pay more than the headphones are worth to have a professional fix them, which means it's unlikely a pro will have the tool and connector you'd need).
If they're quality over-the-ear phones, it may be worth replacing the entire cable, but otherwise, it will be more cost-effective to replace the complete headphone set.
You can crimp the cables and isolate with rubber tape, or in case of "need it right now don't care about tomorrow" you could rip the cables apart some more and manually tie them (always the matching ones of course, color to color). It will look like hell and be not stable. The proper solution would be to replace the cable completely, soldering the end to the speakers. Soldering the broken cable itself will work, but it won't be as flexible at that point anymore.