How is a music speaker able to play more than one note at a time?

By constantly receiving input on how to move, and being able to respond immediately and accurately to any such signal.

Microphones and speakers are similar. Just as a microphone responds to any kind of sound that it can, and a recording device records all the sound that comes in, no matter what it is (to the best of its ability), a playing device plays that same information, and a speaker attempts to reproduce it by vibrating just like the recording device did when it recorded the sound.

Nothing about that is limited to playing one note at a time. In fact, it isn’t even about notes at all. It’s reproducing the sounds that were recorded, whatever they were at the location of the microphone(s).

Music is just complex vibrations in the air. The loudspeaker diaphragm is designed to vibrate and reproduce those vibrations. A very complex pattern of vibration is often required and the surprising thing is that the human ear can pick out the notes of a flute or an oboe from the general cacophony.

Imagine a sound wave for a single frequency. As a function of time, the sound wave forms a sine curve, and a speaker can reproduce it by moving back and forth according to the height of the sine curve. Music is a combination of many frequencies, and you can add their associated sine waves together to form a single complex wave. The speaker now moves back and forth following the complex wave to play the music.

It’s not just the magnets, but the speaker cone that produces sound. The larger the cone, the more notes that it can produce at once since different parts of the cone vibrate differently.

The speaker isn’t playing notes.
It’s playing a sound that is the superposition (add ‘em together) of all the individual sounds in the environment. That can be a combination of all the notes played by a symphony orchestra, or all the voices of kids in the playground, etc.

As @LostInParadise and @dabbler state, the notes are added together (ref: http://spiff.rit.edu/classes/phys207/lectures/beats/add_beats.html). The speaker is simply playing the resulting sum.