The real answer has nothing to do with how speakers work, but the fact that sound waves add in air. When you are listening to a live performance with a singer and instruments, you hear them all together then too. Your question about how a speaker can produce such a composite sound is no different from asking how your ear can hear more than one sound at a time.
Think of what a sound wave really is, which is small aternating pressure variations in the air. You can have multiple sound source, but at any one point in the air you have a single function of pressure over time. This pressure function contains the sum of all the pressures variations caused by all the sound sources.
This varying pressure is what your ears measure at two different points in space (since you have two ears). This sound pressure variation is also what a microphone measures and converters to a electrical signal. At any instant, there is a single pressure value, or a single voltage value coming from a microphone.
A speaker simply does the reverse. It receives a single voltage value at any instant, and produces a local air pressure variation accordingly. Those air pressure variations are ideally the same as measured by the microphone when the recording was made. Since the single-valued function of pressure over time contained the sum of all the sound sources that the microphone picked up, the speaker reproduces the same sum of signals. Your ears then hear the same sum of signals, which allows you to hear the singer and multiple instruments at the same time, just as you would if you were there when the music was performed live.