Translation:The boy has an apple; he is eating it.
I was going to (sarcastically) suggest that the function of á was to fulfil some rule, but that I had no idea what the function of the rule was.
But then it struck me that the subject of the verbal noun is in the genitive, but you can't put é into the genitive. Just as Cat an fhir becomes a chat when you replace the noun "the man's" with the pronoun "his", Tá sé ag ithe úll becomes tá sé á ithe when you replace the noun "apple" with the pronoun "it".
I don't have any source to back this up, but it actually makes a weird kind of sense.
It's completely correct, that is the explanation.
Tá sé á ithe = He is at its eating = He is eating it
Tá sí á h-at féin suas = She is at her own puffing up = She is puffing herself up
(Here you see the typical "h-" for feminine possession)
It really stems from the fact that the verbal noun is a noun.
Tá sé á dhéanamh is really "He is at its making".
Every verbal noun is still a valid noun:
Tá déanamh breá ag na bróga sin.
This is a general rule across the language that things that take the genitive as their object, use the possessive for pronominal objects.
Normally you have ag <verbal-noun> <noun>, and the noun is in the genitive. If you replace the noun with a pronoun, you restructure the phrase to á <verbal-noun>. If the pronoun was é, then you lenite the verbal noun, if the pronoun was í, you don't do anything, and if the pronoun was iad, you eclipse. If the pronoun was mé, the structure is do mo <verbal-noun>
These are the same patterns that you use for the possessive a, and the same goes for verbal nouns that start with vowels - masculine doesn't do anything, feminine gets a h-prefix and plural gets an n-prefix.