What does breaking the fetter 'Clinging to rites and rituals' actually mean?

I've often seen this sort of clinging interpreted as focusing on the rite or ritual itself rather than deeply experiencing the underlying insight or purpose, sometimes intended as a shortcut to doing the work. Examples abound in all religions actually, whereby people think that doing this or that act or repeating such and such a phrase will get you the desired result. Capturing the shell of something, playing the part, but not understanding or abiding in the core meaning. I recall one author writing to the effect that we must see rites and rituals as a means to an end and not mistake them as the ends in themselves.

So how do we know if we're clinging to a ritual? My feeling is that mindful examination is our best tool. Asking ourselves, "Am I just going through the motions with this ritual, perhaps sticking to it merely out of a sense of duty or so that I feel I put in the minimum effort? Or is it truly helping me achieve insight or deepen my practice?"

I do think we all have our little rituals that we indulge in our daily lives, but I would differentiate between a neutral routine (e.g. cup of morning tea) and a ritual that we rely on for personal advancement but we may not be applying ourselves fully to.

The coarser or more general context:

It's about not being attached to symbolic rites and rituals or taking them to be literally true.

For e.g. from SN 7.21:

The Buddha said to Sangarava:

“Is it really true, brahmin, that you practice purification by water, believing in purification by water; that you live committed to the practice of immersing yourself in water at dawn and dusk?”

“Yes, Master Gotama.”

“But brahmin, for what reason do you practice purification by water?”

“It’s because, Master Gotama, whatever bad deeds I’ve done during the day I wash off by bathing at dusk; and whatever bad deeds I’ve done during the night, I wash off by bathing at dawn. That’s the reason why I practice purification by water.”

“The teaching is a lake with shores of ethics, brahmin,
unclouded, praised by the fine to the good.
There the knowledge-masters go to bathe,
and cross to the far shore without getting wet.”

When he had spoken, Saṅgārava said to the Buddha:

“Excellent, Master Gotama! Excellent! … From this day forth, may Master Gotama remember me as a lay follower who has gone for refuge for life.”

The more refined context:

From AN 3.78:

“Take the case of someone who cultivates precepts and observances, a lifestyle, and a spiritual path, taking this as the essence. If unskillful qualities grow while skillful qualities decline, that’s not fruitful. However, if unskillful qualities decline while skillful qualities grow, that is fruitful.”

From MN 24:

“Is the spiritual life lived under the Buddha for the sake of purification of ethics?”

“Certainly not.” ....

“In the same way, reverend, purification of ethics is only for the sake of purification of mind. Purification of mind is only for the sake of purification of view. Purification of view is only for the sake of purification through overcoming doubt. Purification through overcoming doubt is only for the sake of purification of knowledge and vision of the variety of paths. Purification of knowledge and vision of the variety of paths is only for the sake of purification of knowledge and vision of the practice. Purification of knowledge and vision of the practice is only for the sake of purification of knowledge and vision. Purification of knowledge and vision is only for the sake of extinguishment by not grasping. The spiritual life is lived under the Buddha for the sake of extinguishment by not grasping.”

From DN 16:

If it wishes, after my passing the Saṅgha may abolish the lesser and minor training rules.

The purpose of precepts and observances, purification of ethics, and a spiritual path is only for the purification of the mind and cultivation of skillful qualities.

But being too attached to minor rules and rituals to the point that breaking them causes remorse and mental anguish, is unskillful and pollutes the mind.

It is enough to practise purification of ethics with the understanding that it is for the purification of the mind, and not be too attached to following minor rules and rituals to the letter.