إِنِّي combines inna with a first-person pronoun, conveying “indeed I.” صَائِمٌ states the speaker’s condition of fasting. At this stage, recognizing the meaning of the complete clause matters more than memorizing a full grammatical analysis.
In وَهُوَ يَتَغَدَّى, هُوَ is a pronoun and يَتَغَدَّى is a present-tense verb. Together they describe the Prophet’s ﷺ condition when Bilal entered.
In مَا أُكِلَ عِنْدَهُ, عِنْدَهُ means “in his presence.” It does not mean that the fasting person himself was eating.