I had to cut the wires and connect the copper strands together, I basically just twisted it together and put electrical tape to hold it. My question is that I cut some of the copper strands of wire when trying to peel off the protective rubber on the wire. Will that affect the performance of the speaker and should I open it up and redo it?
Most of the speaker wires have more strands or bigger wire diameter than it requires.
But sometimes there is cheaper speaker wire with less strand than it should be so when you cut let's say 3 strands of it, it could make a difference.
Redo the process with the better and more precise procedure is better than you left at it and still thinking it could be better :)
If you think the speaker wire diameter is too small, you could replace the wire with a bigger gauge. (i mean bigger diameter because bigger gauge = less diameter)
For calculation and standard size of speaker wire / watt you can check in this link:
I've made that mistake countless times over 35 years and never had it affect electrical conductivity. You'd have to cut away like more than half the strands before you'd have a problem, or if the speaker is very powerful and draws high amperage.
What is much more of a concern is the use of electrical tape alone. The tape tends to unravel when wrapped around a thin wire, unless you melt the tape a little with a cigarette lighter and squeeze it. I have had wires jiggle themselves loose due to vibration inside the tape, causing tiny amounts of arcing and poor performance. I've also had problems with corrosion in dry twists, especially if the wire is copper.
Next time, use insulated wire connectors that you crimp with pliers.