There are two ways of tackling this. You can either switch the low-voltage DC power to the EL wire inverter (which is safe and easy) or you can switch the high voltage AC power out of the inverter (which is harder, more dangerous, but does allow you to route the power to different EL wires for animation).
The former can be arranged in just the same way as switching any heavy load from an Arduino - with an N-channel MOSFET as a low-sided switch, or a combination of N- and P-channel MOSFETs as a high-side switch. You could add further isolation with a relay should you wish. BJTs could be used instead of MOSFETs but MOSFETs are better at switching higher currents from low current sources.
For switching the high-voltage side you will need a TRIAC circuit. While it is possible to switch a TRIAC direct from the Arduino it is preferred to opt-isolate the system from the Arduino both for safety and ease of switching at higher voltages.
The most common opto-isolator to use is the MOC3041 Opto-Isolated Triac. This is used to switch the gate of a higher voltage / power TRIAC to switch the load.
In that circuit the "220V" comes from the EL inverter, and the LOAD is the EL wire itself. Each EL wire you want to switch would use a copy of that circuit, but can (power permitting) use the same inverter as the source.