Among the special priestly garments worn by the Kohen Gadol is the Me’il, or Robe:
And you shall make the Robe (Me’il) [worn under] the Ephod, entirely of blue wool. And its top opening shall be inwards; its opening shall have a border all around of woven work; like the opening of a coat of mail shall it have; it shall not be torn. And you shall make on its hem pomegranates of blue, purple and scarlet wool for its hem all around, and golden bells in the midst of them all around: a golden bell and a pomegranate, a golden bell and a pomegranate, all around the lower edge of the Robe. And it shall be [worn] on Aharon to minister; and its sound shall be heard when he comes to the Holy, before Hashem, and when he goes out, so that he shall not die
(Shemot 28:31-35).

The Robe (Me’il) was long enough to come down to the feet of the Kohen Gadol. The opening for his head was part of the weave, preventing tearing. All along the bottom hem, at the feet of the Kohen Gadol, the Robe (Me’il) was decorated with a series of “pomegranates” made of blue, purple and scarlet wool, and bells. Rashi says (28:33) the “pomegranates” were
“spherical and hollow in the shape of pomegranates, made like chickens’ eggs.”

According to Rashi’s interpretation, the “pomegranates” and golden bells alternated all around the bottom of the Me’il.

These pomegranates and bells are an indispensable part of the design of the Me’il. They are not only decorative; they are functional:
and its sound shall be heard when he comes to the Holy, before Hashem, and when he goes out…
Moreover, the sound of the bells prevent the death of the Kohen Gadol:
…so that he shall not die.
From this Rashi (28:35) deduces a law for all the priestly vestments:
“From the negative one can derive the positive: if he will have them he will not be liable for death; thus, if he enters lacking one of these garments he is liable for death at the hands of Heaven.”
Rashi’s sources are Sanhedrin 83a and Zevachim 17b; Rambam (“Laws of the Sanctuary Vessels” 10:4) rules that this is the law.

Why must the sound of the bells be heard? And by whom?

Ramban points out that it is not usual for such splendid clothes as are worn by the Kohen Gadol to be decorated with bells. And of course Hashem does not need to be informed of the coming of the Kohen Gadol into the Sanctuary. Nevertheless, the sound of the bells is needed so that
“He might enter before his Master as though with permission, because one who enters the chamber of the King suddenly deserves death according to the protocols of sovereignty.”
This is comparable to the dicta of mortal kings, as in the case of Achashveirosh:
Any man or woman who comes before the king to the inner court without having been summoned, there is but one law for him, to be executed
(Esther 4:11).
Similarly, the sounding of the Me’il bells is a request for permission
when he comes to the Holy, before Hashem, …
And permission is also requested before leaving the Presence of the King:
and when he goes out.

In addition, explains Rabbenu Bachya (ben Asher ben Hlava, 13th Century) based on Ramban,
“It is a warning to the angels that they clear the way for the King’s close friend [the Kohen Gadol] in honor of the King, so that he can enter and serve Him in private. Also, so that the Kohen will not be harmed if he comes in suddenly … because the attendants [angels] who are there around the Divine Presence will attack him.”
Hashem is ministered by His angelic hosts; still, He desires that His earthly servants worship Him. The jealous angels are bidden by the bells of the Me’il to make way for Hashem’s preferred human ministers, the Jewish People, as led by the Kohen Gadol.

R. Shmuel Alter (20th Century) in Likutei Batar Likutei, derives from the Me’il bells a message to Aharon, and indeed to every community leader. He must first know that leadership is not mastery over his people, but service to them:
And it shall be on Aharon to minister.
When he has internalized this idea, and projects it through his all words and deeds, then and only then will he be heeded by his people:
and its sound [KOLO, literally, his voice] shall be heard.
Of course, many leaders are accepted by their people when they are first inaugurated:
when he comes to the Holy;
but the true indication that the leader is followed comes when he has completed his term:
and when he goes out.
And then his fame and good works will endure forever:
he shall not die.

Our Sages derive from the example of the Me’il bells another important lesson in proper interpersonal behavior. Rabbi Akiva instructed his son R. Yehoshua that, among the seven ways a true talmid chacham can be recognized is
“Do not enter your house ― and certainly not your friend’s house ― suddenly (Pesachim 112a).”
In order to show sensitivity to the privacy of everyone in the house, one must announce his coming by ringing the bell or knocking on the door. Furthermore,
“When R. Yochanan would enter his house he would make a shaking sound, because it says, and its sound shall be heard when he comes to the Holy (Vayikra Rabba 21).”
R. Yochanan applies the term the Holy to his own home! Our homes – especially in the absence of a Temple – are the places where we imbibe sanctity, where our every action is a way to serve Hashem.

The bells of the Me’il are meant to be heard by all – by Hashem and by His angels, as well as by every community leader and everyone in our holy homes.