The vibration of some physical object can cause the air to vibrate. The transmission of physical vibrations of objects to motions of the air is complex and varies by instrument.
In a string instrument, a player plucks or bows the string which vibrates with a characteristic frequency. A string all by itself doesn't sound like much. The bridge, the sound post, and the body of the instrument act to transmit the motion of the string into motions of the air. The size, shape, and thickness of the parts of the body of a string instrument serve to 'couple' the sound to the air. Even the size, shape, and placement of the holes in the body is critical making the instrument sound the way it should.
In a piano, there is a sound board which serves the function of transmitting the vibrations of the strings to vibrations in the air.
In brass instruments, the motion of the lips opening and closing the airway in rapid succession causes the air column inside the instrument to resonate. This is nearly identical to what happens in reed instruments which use a single or double reed to open and close the airway. For many of these instruments the bell shape of the end is required to get some of the sound out of the instrument. If the tube simply ended without a flare, very little of the sound energy resonating inside would be able to reach our ears.
Flutes, organ pipes, and whistles are similar, but rather than the opening and closing of the air column, turbulent flow of air across the hole drives the resonance of the air column.
Percussion instruments are often a little more direct in converting their motion to sound. A drum head transmits vibrations into the air easily. The same is true for many bells and cymbals. But marimbas have tubes like organ pipes which help to convert the motion of the bars into sound.