The reeds vibrate, but interestingly it is not the reed vibration that causes the harmonica sound. You can verify this for yourself by removing the covers and plucking a reed. It makes a tiny ping sound, nothing like what you hear when you play the harmonica. The sound of the harmonica is created by the action of the reed vibrating through its slot and chopping the air stream. This is similar to the way a siren works. There is very little clearance between the edges of the reeds and the sides of the slots in the reed plates, so when the reed is actually within the slot during its vibrations the air stream is essentially shut off. When the reed emerges from its slot on either side of the reed plate the area through which the air stream can pass opens up, so the air stream starts up again. It is this vibration of the air stream that makes the harmonica sound. The reeds are not like guitar strings, which vibrate to make the sound which is amplified by the acoustics of the guitar body. They are just springs that cause vibrations in the air stream.
The tiny harmonica does not have any resonant body like a guitar or violin, or a sound board like a piano. Nothing in the harp appreciably amplifies its sound or resonates. The resonance comes from the player, which is largely why the harmonica is such a personal instrument. Many people will tell you that the material out of which the body of the harmonica is made determines how it will sound. While there may be very small differences in tone due to materials, any such difference is essentially negligible to all but the most advanced players.
The top picture above shows the harmonica parts in their proper orientation. Notice how from this angle you can only see the slots in the reed plates on the other side of the plate from where the reeds are attached. Both plates have their reeds facing down. You can see that when the harp is assembled, the upper reeds are inside the comb and the lower reeds are outside the comb.
Normal blow notes are caused by the upper reeds crossing their slots when the air stream enters through the holes in the comb, flows over the inner reeds, and exits through the slots. The reeds move into their slots, stopping the air stream, and then out the other side of the reed plate, which lets the air flow again. As the reed deflects it gains energy like stretching a spring or a rubber band. When the air pressure is released after the reed passes through its slot the energy in the reed causes it to spring back toward its starting position, once again crossing its slot and cutting off the air stream. This process repeats while the air stream is maintained. Normal draw notes work similarly but are caused by the lower reeds crossing their slots when the air stream flows over the outer reeds, enters through the slots, and exits through the holes in the comb. This action of the reeds where they first move into their slots is traditionally called a closing note. Bends to change the pitch of closing notes are called closing bends.
During normal draw or blow bends, both the draw reed and the blow reed can participate in making the note. As you start with a draw note and bend it down to its limit, initially the draw reed makes the note as above, then both reeds vibrate through their slots to make the note, and finally, at the deepest part of the bend, only the blow reed is making the note. You can verify this by removing the covers and using your fingers to stop the vibration of the upper and lower reeds at various times during a bend. The reverse happens during blow bends: the blow reed starts, both reeds participate, and finally only the draw reed makes the sound.
The range of bending available for a pair of reeds is determined by the pitches of the natural notes of the reeds (i.e. the unbent notes). The pitch can be bend down from the high note to just lower than 1 semitone (one half step, e.g. B to C) above the low note.
The opposing reed, i.e. the blow reed during a draw bend, starts by moving away from its slot. This deflection adds energy to the spring that is the reed. When resonance factors are just right the opposing reed can gain enough energy that when it springs back it goes all the way through its slot. The action of the reeds where they first move away from their slots is traditionally called an opening note. The physics of exactly how all this interplay between the reeds, air stream, and slots works is largely unexplored, unintuitive, and not well understood.
The point to understand is that during a normal bend, one reed is operating in a closing note fashion while the opposing reed is operating in an opening note fashion. The closing reed's pitch gets lower while the opening reed's pitch gets higher than their corresponding natural notes.
During overbends, i.e. overblows and overdraws, the resonances and air flow characteristics are such that the closing reeds don't participate in making the sound, and only the opening note is played. (It is possible to play in such a way that the closing and opening reeds both play and produce different pitches. This is usually caused by poor overblow technique, but can be used to achieve two-note double stops that can't be achieved any other way.) The action of stopping the closing reed from sounding is often called choking the reed, which forces the closing reed into its slot with very little energy so that it neither crosses its slot nor has enough energy to spring all the way back to its normal position. With the slot blocked the air stream cannot flow that way, and with the little energy in the spring the reed remains in position to block the slot, so the closing reed vibration never takes place. Overbends are opening notes, so their pitch is higher than the natural note of the reed--just the opposite of normal bends.
How does the harmonica work? You inhale through it or exhale through
it and it makes notes. You change the shape of your mouth, move your
tongue around, open and close your throat, breathe with different pressures
and attacks, and the notes it makes changes. You do it enough and
you figure out what makes what happen.