The verse in Ex 28:33 is just one of several on the same subject:
- Ex 28:35 - Aaron must wear the robe whenever he ministers, and its sound will be heard when he enters or exits the sanctuary before the LORD, so that he will not die.
- Lev 16:2 - And the LORD said to Moses: “Tell your brother Aaron not to enter freely into the Most Holy Place behind the veil in front of the mercy seat on the ark, or else he will die, because I appear in the cloud above the mercy seat.
- Lev 16:13 - He is to put the incense on the fire before the LORD, and the cloud of incense will cover the mercy seat above the Testimony, so that he will not die.
There is a Jewish tradition about this explained by Got Questions, in https://www.gotquestions.org/high-priest-rope.html from which I now quote:
... God warns the high priest to follow the protocol, on penalty of death (verses 2 and 13). It’s possible that, because of the seriousness of entering the Holy of Holies, the Jews formulated the idea of tying a rope around the high priest’s foot.
According to tradition, during the last couple of centuries of the temple’s existence, a gold or scarlet rope was tied to the high priest’s foot on the Day of Atonement. Another priest standing in the Holy Place tended the other end of this rope. If the high priest’s sins were not atoned for properly, then God would strike him dead when he entered the presence of the Shekinah—the glory of God—that filled the Holy of Holies. Since access to that part of the temple was strictly forbidden, the priests felt they needed a way to retrieve the body of the high priest, if necessary.
Exodus 28:33–35 specified that bells (and knitted pomegranates) were to be worn on the hem of the high priest’s garment so that “the sound of the bells will be heard when he enters the Holy Place before the Lord and when he comes out” (verse 35). The bells provided an audible cue to those outside the tabernacle of the high priest’s whereabouts. They could tell when he entered and exited the Holy of Holies. According to the rope tradition, the bells also provided a means of knowing if the priest had died: if the jingling stopped, the priest on the other end of the rope would be alerted to the fact that the priest had ceased moving. There is nothing in Scripture to indicate that was the purpose of the bells.
I also note this about the bells on the High Priest's robes quoted from Ellicott when commenting on Ex 28:33 -
Bells of gold.—The bell is also more Assyrian than Egyptian. Its use as an article of priestly costume has no direct parallel, nor are bells known to have been employed in the religious services of any ancient nation. The statement that Persian kings wore bells rests upon no sufficient authority. We seem to have here the introduction of an entirely new religious usage.
Therefore, apart from the dubious Jewish tradition explained above, we do not know the purpose of the High priest's bells. At the same time, while there is no Biblical basis to confirm the above tradition, there is also nothing in the bible that prevents that tradition perhaps being true.