Here's my personal interpretation of SN 12.63.
To understand each of the similes, the clue is in the epilogue of each simile. The nutriments are not taken literally. Each simile becomes harder to interpret than the previous one. Each simile is also related in some way to the previous simile.
The epilogue says "When the nutriment edible food is fully understood, lust for the five cords of sensual pleasure is fully understood." The five cords of sensual pleasure (MN13) are forms, sounds, aroma, flavours and tactile sensations.
The nutriment of edible food here is not literally edible food, but refers to the external sources of forms, sounds, aroma, flavours and tactile sensations.
The simile teaches us that these "edible food" should be used only as a means to get through samsara, while living the holy life, to reach the end of suffering, and one should not have passion for them.
The nutriment of the external sources of the five cords of sensual pleasure, nourish and sustain the five cords of sensual pleasure. Literal edible food is the source of flavours, for example.
The epilogue says "When the nutriment contact is fully understood, the three kinds of feeling are fully understood." The three types of feeling (Iti 52) are pleasure, pain and neither-pleasure-nor-pain. There are six types of contacts (SN 12.2) - eye-contact, ear-contact, nose-contact, tongue-contact, body-contact and intellect-contact.
The cow is exposed to different environments like wall, tree, water, open air. These are like contacts. The creatures biting her are the three types of feeling. The biting itself is craving. The painful sensation that arises from the biting is suffering (from dependent origination: from feeling as a requisite condition comes craving; and from the second noble truth: the cause of suffering is craving).
The simile teaches us to abandon passion for the three types of feeling arising from contacts. The creatures (feeling) are fine, but don't let them bite you (craving). For the bite is painful (suffering). The nutriment of contacts nourish and sustain the three types of feeling.
To link back to the previous simile, SN 12.44 says for example, "dependent on the ear & sounds there arises ear-consciousness. The meeting of the three is contact". So, five types of contacts are directly related to the five cords of sensual pleasure. The sixth type of contact is indirectly related to the five cords of sensual pleasure.
The epilogue says "When the nutriment mental volition is fully understood, the three kinds of craving are fully understood." The three types of craving (Iti 58) are craving for sensuality, craving for becoming and craving for non-becoming.
The word for "nutriment of mental volition" in this sutta is manosañcetanāhāro. From SN 12.38, we read: "what one intends, and what one plans, and whatever one has a tendency towards: this becomes a basis for the maintenance of consciousness." I interpret mental volition here as intention (cetana) and latent tendencies (anusaya).
SN 12.38 also says "If one does not intend, and one does not plan, but one still has a tendency towards something, this becomes a basis for the maintenance of consciousness", which shows latent tendencies are required for the maintenance of consciousness, even if intention isn't present. There are 7 types of latent tendencies in AN 7.11.
The two strong men are either latent tendencies (anusaya) or latent tendencies and intention (cetana). They grab the man and drag him towards the charcoal pit with glowing coals. The glowing coals are craving. If the man comes into contact with the glowing coals, the arising pain is suffering (from the second noble truth: the cause of suffering is craving).
The simile teaches us to abandon latent tendencies. Don't let the strong men (latent tendencies) drag you into the charcoal pit of glowing coals (craving). Run away from the strong men (latent tendencies). The coals are painful (suffering). The nutriment of latent tendencies nourish and sustain the three types of craving.
To link back to the previous simile, the three types of feeling will not result in the arising of craving, if there are no latent tendencies.
The epilogue says "When the nutriment consciousness is fully understood, name-and-form is fully understood." Based on the definition of name-and-form in SN 12.2, name-and-form is a kind of mind-body link (according to Theravada interpretation).
And what is consciousness? From dependent origination: from consciousness as a requisite condition comes name-and-form. From the traditional Theravada interpretation of consciousness in dependent origination, this is the consciousness that results in the continuity of suffering beyond physical death, called "rebirth linking consciousness" (cuti citta) in the Abhidhamma.
This is supported by SN 12.39: "When consciousness is established and has come to growth, there is a descent of name-and-form", and SN 12.38: "When consciousness is established and has come to growth, there is the production of future renewed existence." When the suttas talk about the growth, increase, maturing, landing and establishing of consciousness, it essentially talks about consciousness resulting in the descent of name-and-form ("rebirth").
In dependent origination, craving leads to clinging, which leads to becoming. These show the outflow of effluents or mental fermentations or taints (asava), which according to this "cyclic" question causes the arising of ignorance. Then ignorance in turn causes the arising of effluents once again (MN 9), which is mental formations (sankhara) including latent tendencies (anusaya). This then causes the arising of consciousness, which causes the descent of name-and-form (mind-body link) i.e. "rebirth".
This loop back from craving-clinging-becoming back to ignorance is hinted at, in SN 44.9: "when a being sets this body aside and is not yet reborn in another body, I designate it as craving-sustained, for craving is its sustenance at that time."
The bandit or criminal is the one who delights in craving, the fool from SN 12.19, who clings (steals). The king could be ignorance or samsara. The king's men who strike him with spears are mental formations (especially latent tendencies). The spears are (rebirth linking) consciousness. The striking of the spears is the descent of name-and-form (that is "rebirth"). The pain caused by the striking of spears is suffering (birth, aging, death, sorrow, lamentation, pain, distress and despair). Morning, afternoon and evening could be different aeons.
Wouldn't three hundred lifetimes be painful? Well, even one lifetime is painful enough, don't you find from experience? The simile teaches us to abandon stealing (craving), in order to not get struck ("rebirth") by spears (renewed "rebirth linking consciousness") again.
This is reinforced in the Tears Sutta (SN 15.3):
From an inconstruable beginning comes transmigration. A beginning
point is not evident, though beings hindered by ignorance and fettered
by craving are transmigrating & wandering on. Long have you thus
experienced stress, experienced pain, experienced loss, swelling the
cemeteries — enough to become disenchanted with all fabricated things,
enough to become dispassionate, enough to be released."
The nutriment of (rebirth linking) consciousness nourishes and sustains the (descent of) name-and-form.
To link back to the previous simile, the three types of craving (plus clinging and becoming) are the effluents that cause ignorance, and ignorance causes effluents (latent tendencies especially) in turn, that causes (rebirth linking) consciousness to rise again, causing the descent of name-and-form ("rebirth").
Case of arahants
How does this apply to the case of arahants?
From Thanissaro Bhikkhu's commentary of Iti 44:
With fuel remaining (sa-upadisesa) and with no fuel remaining
(anupadisesa): The analogy here is to a fire. In the first case, the
flames are out, but the embers are still glowing. In the second, the
fire is so thoroughly out that the embers have grown cold. The "fuel"
here is the five aggregates. While the arahant is
still alive, he/she still experiences the five aggregates, but they do
not burn with the fires of passion, aversion, or delusion. When the
arahant passes away, there is no longer any experience of aggregates
here or anywhere else.
A living arahant experiences the nutriment of edible food and the nutriment of contacts, but he has ended the nutriment of mental volition (i.e. latent tendencies) and therefore also ended the nutriment of (rebirth linking) consciousness. Then, consciousness has no place to land or get established any more (SN 12.64) i.e. no more descent of name-and-form (SN 12.39) meaning no more continuity of suffering after physical death. After parinibbana, all nutriments would have ceased permanently.
The sutta is talking about dependent origination, what keeps dependent origination going, how to end dependent origination, and the truth of the continuity of suffering beyond physical death.
It is true then that the noble eightfold path does not just lead to the cessation of (craving for) form, but also the cessation of (the rearising of) form. (see this question)