Music plays a huge role in Aztec (and other central American) cultures. As such, its meaning is complex, and a lot has been written about both Aztec music in general and the specific meaning of the breaking of the flutes.
The most basic interpretation is that the breaking of the flutes simply represents the death of the sacrificial victim/avatar of the god on earth:
What then do the flutes, flute playing, and the physical breaking of the flutes by the player himself signify? Not only is a painful death the end of the happiness of earth, but the flute also symbolizes all that is humanly and spiritually powerful, as Carrasco explains: "The sound of the flute, as with the word, depends on the exhalation of the breath or, in Nahua thought, the soul." Therefore, the soul of the sacrificial victim who represented the "god of gods" is forever lost with his death and by his own breaking of the flutes that represented his soul.
(Olsen, Dale A., World Flutelore: Folktales, Myths, and Other Stories of Magical Flute Power.)
This is further supported by the fact that flutes were a metaphor for flowers, meaning that the act of breaking the flute was analogous to picking a flower (i.e. killing them, but preserving their "essence" like preserving a flower in a vase):
Furthermore, the particular Aztec ceramic duct flute... performed by the young impersonator of Tezcatlipoca is called a "flower flute" because it has a flared distal end that represents a flower blossom. Furthermore, the flower... symbolizes beauty, blood offering, and music, according to Arnd Adje Both:
Indeed, the heart of a deity impersonator itself was compared to a flower, metaphorically "plucked" and offered in ritual human sacrifice. Analogously the impersonator "plucked" the flutes before his own death, as symbolized in breaking the bells from the tube over the stairway and leaving them as undamaged as possible. This act of ritual destruction is clearly evident by organological data and could be interpreted as a process of transformation into the spiritual realm.
Here are some pictures of the flower shaped flutes used in the ritual (from Arnd Adje Both's fantastic pdf about the instruments).
Another interpretation focuses on the use of music to communicate with the gods. In fact:
Some musical instruments were considered to be sounding idols, indicating that ritual music could be perceived as a voice of the gods, and that the Aztec instrumentalist fulfilled the role of an expert mediator through whom a god sang.
(Both, Arnd Adje, Aztec Music Culture.)
The importance of music is further emphasized by the myths that discuss the origin of music. In Central American myths, music is stolen from gods (often with the help of Tezcatlipoca, and gives humans the power to communicate with the spiritual realm:
An interesting myth... related that originally music was the property of the sun. After the sacrifice of the gods in Teotihuacan, men gathered their remains and made sacred bundles out of them, but they did not know how to adore them. Tezcatlipoca created Ehecatl (Wind), whom he entrusted with the task of going to the house of the sun to bring the musicians there to the earth. Water goddesses, the nieces of Tezcatlipoca, build a bridge [sic] so that Ehecatl could reach the abode of the sun. The musicians were warned not to answer the intruder with the "mellifluous song," but in spite of these warnings, one of the musicians succumbed to the charm of the song and followed the envoy of Tezcatlipoca. That was the origin of music and prayers on the earth, thanks to which men could adore the gods.
In the Popol Vuh, the noise of the ball game provokes the ire of the underworld masters of Xibalba; several modern myths give the same function to the song and music played by the hero... Thus the noise from the ball game, smoke, and music all have parallel functions. They manifest the acquisition of cultural goods, which constitute a king of usurpation of the power of the creator gods.
(Oliver, Guilhem, The Hidden King and the Broken Flutes: the Mythical and Royal Dimension of Tezcatlipoca's Feast in Toxcatl.)
This is also supported by the Aztec connection between flowers and music (which we've previously discussed when comparing the shape of the flutes to flowers). The scent of flowers and the sound of music are analogous in Aztec culture, as both allow humans to communicate with their deities.
The Aztec concept of music as a “flowery song” is reflected in the calyx-shape of the flower-flute. To play it allows to visualize the metaphorical link between sound and the scent of blossoms, both regarded as sacrificial gifts functioning as a means of communication with the spiritual realm, probably because of their invisible sensual quality. That the ruler was compared to the flute, through which Tezcatlipoca could express his will, makes the important social function of the instrument evident and shows that instrumental sound was perceived as the proper voice or ‘flowery song’ of a deity. Thus, the flutist fulfilled the role of a mediator. It can be proposed that he was sacrificed instead of the ruler.
(Both, Arnd Adje. Aztec Flower-Flutes: The Symbolic Organization of Sound in Late Postclassic Mesoamerica.)
Therefore, breaking of the flutes represents a pause in communication with the gods, which is then reinstated after the sacrifice occurs. (I think it's also reasonable to interpret the breaking of the flutes as briefly returning the gift of music to the gods, and thereby thanking them for said gift.)
Recall that in the myth the envoy-counterpart of Tezcatlipoca descended from the abode of the sun and offered music to the people so that they could adore their gods. In the ritual, the representative of Tezcatlipoca returned toward the sun [i.e. walked back up the temple] and broke the flutes. By breaking them on the temple steps, he foresaw his own death by becoming one with the instrument. With his death the possibility of contact with the god was broken, and only with the coming of a new representative would it be possible to reestablish a link between gods and humans. The king could then leave his retreat, and the sound of flutes would again make the presence of the god known in the city.
Yet another interpretation looks at the ritual through the lens of Aztec and Central American ideas about the soul and human nature.
The sacrifice of the youth was linked to the profound philosophical idea that only the true, the deified heart is worthy to become nourishment for the great start that maintains life on earth. The Nahua peoples believed that we are born with a physical heart and face, but that we have to create a deified heart and a true face.
(Nicholson, Irene. Mexican and Central American Mythology.)
The ceremony was a myth acted out in real life. The myth was about man, who must learn to glorify god through sensual things, through fine clothes and music and dance, before he is worthy of breaking the senses one by one and losing his life in order to gain it.
This interpretation of the breaking of the flutes reminds me of the following quote about art:
The poet and critic Allen Grossman tells a story (there are many versions of the story) that goes like this: you’re moved to write a poem because of some transcendent impulse to get beyond the human, the historical, the finite. But as soon as you move from that impulse to the actual poem, the song of the infinite is compromised by the finitude of its terms. So the poem is always a record of failure.
(Lerner, Ben. On Disliking Poetry)
And now we've come full circle: a reason why flutes are so important is that music allows humans to express themselves. Breaking a flute is the equivalent to killing someone as they can not speak nor communicate with the gods. However, the breaking of the flute also represents both an interruption of divine communication and (as argued by Nicholson) an attempt at a rebirth into a realm where flutes aren't necessary, because there is direct communication with the spiritual realm.
Of course, none of that theology makes the human sacrifices performed by the Aztecs ethical, but it helps us understand their perspective better.