According to the Anapanasati Sutta it is this practise that leads to liberation but how does the destruction of (a) fetter(s) exactly happen? Let's take the stage from once-returner to non-returner. The latter one has removed sensual desire and anger. How did this exactly happen during Anapanasati?
I think the progression described in the Anapanasati Sutta can only lead to stream-entry and once-returner but not to non-returner or Arahant because the Anapanasati Sutta includes knowing of in & out breathing at every stage therefore does not yet include the jhana (that has no knowing of breathing) required for non-returner or Arahant.
This said, both the stream-enterer and particularly the once-returner have significantly reduced sensual desire and anger.
How this exactly happens during Anapanasati is insight (vipassana) into not-self (anatta) reduces the self-centredness that is the root of sensual desire and anger and particularly the calming (samatha) stages (4, 8, 10, 11, 12, 14 & 15) provide the non-sensual happiness that allows the mind to naturally lose interest in sensual pleasures. Once sensual desire reduces, the impulse to worldly anger reduces (although some tendency to anger due to moral righteousness may remain).
Stages 5, 6 & 7 are also important (vipassana) because they allow the mind to transcend or decouple from enslavement to pleasant feelings; when it is discerned the peace of non-attachment & samatha is more pleasant than the strong vibrations of pleasant feelings; particularly the agitating quality of rapture (piti). Naturally losing interest in rapture by intimitately experiencing rapture's true nature is a very significant step to losing interest in sensuality.
Stage 9 is also important, where the ugly polluting nature of defilements are discerned directly; like coming face to face with the devil.
In summary, an important result of Anapansati is calm (samatha). When calm is experienced, it feels better than sensual desire or anger; thus breaking/reducing these fetters. Feeling this calm starts at stage 3, which is actually called: "experiencing all bodies (kaya)". The word 'kaya' refers to the physical body, breath-body and mental-body (nama-kaya). Stage 3 is experiencing how a mind of non-desire, non-anger & non-attachment results in the breathing calming and the physical body calming. Stage 3 is about experiencing the 'inter-relatedness' between the mind, breath & body. At stage 3 alone (when properly practised), it is experienced how a mind without sensual desire & without anger causes the breathing, body & mind to calm & feel satisfying peace.
Anapanasati MN118 is a progression of meditating with breath. This progression is very methodical and it is a guide to relinquishing what we hold on to, a guide to letting go. The progression starts with the breath itself and invites an awareness of the quality of the breath, not a controlling awareness to make the breath anything, but an abiding awareness to observe what the breath is doing. Retaining and extending that awareness as the sutta progresses, one finds this:
Whenever a mendicant practices breathing while experiencing the mind...at that time they meditate observing an aspect of the mind—keen, aware, and mindful, rid of desire and aversion for the world.
As noted by @Dhammadhattu, the phrase rid of desire and aversion for the world occurs in each of the four tetrads, so the release to these happens progressively via each tetrad. Skipping tetrads to achieve a particular result is ineffective. Instead, simply follow the progression. The progression works. Sometimes even the first few steps are enough to deal with daily issues. Relinquishing the fetters is a gradual process. One thread at a time. One day at a time. Fetters are intricate and constant practice attending to breath is most effective. Be thorough. Be diligent.