It is never explained.
Spellcasting normally consists of drawing something in the air (S) with something from the spell-component pouch in your hand (M), while saying something in Latin/Quenya (V) —as shown in the picture.
In Pathfinder there is a feat called Spellsong, which meant one sneakily wove the spellcasting into a musical performance. In that case the musical instrument's music is the Verbal component, the instrument itself the focus (Material), while the Somatic is probably crazy hand-gestures/moves/guitar-stunts (S can be cast with a free hand or the hand holding the focus). To make it par with the other classes, the instrument is somehow played one-handed.
Additionally, at mid level, one does Bardic Performances as swift actions to boost allies. Even though there is no mention of a different instrument, I like to think is it, making the bard a one-man-band —technically the Dissonant Voice feat makes Singing the chosen performance for the Bardic Performance—, while musical instruments with bonuses for the spellsinging as the spellcasting performance is countered by the target's perception check —technically there is ambiguously written somewhere obscure that the bonus mightn't apply. As a result, the bard is a one-man-band that performs magical music.
In D&D 5e, there is no mention of "music" or "note" in any spell description, while the fluff in the description of bard (PHB p.52) says:
- “You have learned to untangle and reshape the fabric of reality in
harmony with your wishes and music.”
- “Your magic comes from the heart and soul you pour into the performance of your music or oration.”
Bardic Inspiration on the other hand says:
- “You can inspire others through stirring words or music.”
However, there is no harm however picturing the bard as playing an instrument or singing (plus spell-component pouch) to cast musical magic. In the case of spells without Verbal (e.g. minor illusion), it could be see as playing pianississimo (very very softly), or a rhythmic tap.
How does one hold a musical instrument one-handed?here), therefore renouncing all physical credibility in favour of awesomeness…