My brain just can't comprehend how all the sounds and frequencies at one single point in a song (drums, vocals, guitar, etc) can be created by one single vibration of a membrane. All at once!? I really need an explanation.
There's quite a few ways this question can be answered, but perhaps the most concise way to do it is to ask you to consider the properties of waves in general. You understand that in order to produce sound, the membrane has to vibrate. In order to vibrate, a force must act upon it and that force is provided by the coil and the magnet that sit at the apex of the membrane. An alternating current is applied to the coil, which - through the action of the electrical and magnetic fields - is translated into waves of kinetic energy, which travel along the membrane and produce the sound waves you hear.
The key point here is - waves. All 'waves' of this nature exhibit the property of 'superposition', which means that one wave can be superimposed on top of another wave without either of the waves affecting each other. You can easily see this by going to a local pool and dropping in a couple of stones then observing how the waves interact.
So in reality, the membrane isn't undergoing just one single vibration, it's undergoing many - superimposed - vibrations - all at the same time. Each of these superimposed vibrations are being translated by the membrane into the sound waves you hear - frequencies from as low as 20Hz all the way up to 16kHz - depending on how much hearing you have left. Higher frequencies are better reproduced by transducers with slightly different physical properties than those that reproduce lower frequencies, which is quite often why you see speakers with multiple 'drivers'.
Unfortunately, your question cannot be answered without getting into lengthy discourse about cabinet design and resonance, so I will leave it here at this point and direct you to papers written by Thiell and Small which give more accurate formulae and algorithms for speaker resonance and design.
However, the short and concise answer to your question is to say that waves exhibit the property of wave 'superposition' which is how many waves and vibrations can be transmitted through a membrane at the same time.
The membrane does not oscillate: it is dampened to the degree that it does not entertain any relevant oscillations of its own. It just moves, following the electrical signal. One consequence is that loudspeakers are much much less efficient in producing sound energy from electrical energy than acoustic instruments are in producing sound energy from mechanical energy. For musical instruments, a significant part of the energy not sustaining an oscillation/vibration is eventually transmitted as sound energy. For loudspeakers, by far the largest part of the energy is converted into heat.
Another consequence is that a loudspeaker does not significantly resonate and can support transmitting an arbitrary overlay of multiple independent frequencies equally well. In contrast, the resonators of an acoustic instruments are very much limited to producing a single frequency and overtones at one point of time.
There are more and less efficient loudspeakers: very good horn speakers excel at transferring a lot of mechanical energy into sound immediately and thus actually have reasonable efficiency. The efficiency of musical instruments is usually rather achieved by having a quite less thorough transfer of sound energy but that doesn't hurt that much since the energy that isn't transferred is mostly preserved in a resonator that stores and accumulates energy at the resonator's current nominal frequency. A loudspeaker's non-transferred energy is instead almost immediately converted into heat by its damping.
A bass reflex box actually has a bit of resonating going on, entertaining standing waves at a low frequency range. That increases the efficiency while at the same time watering down the precision of the reproduction since onset and decay at those frequencies are delayed.