laylatahum: accusative, functioning as an adverbial of time.

a yyuhum: the interrogative noun ayy is read with ḍamma because it is treated as indeclinable through annexation, and the introductory element of its relative clause has been omitted.

ʿAlī ibn Abī Ṭālib: the Prophet’s cousin, husband of his daughter Fāṭimah, the fourth caliph, one of the earliest to accept Islam, and one of the ten promised Paradise; he was assassinated in 40 AH.

yashtakī ʿaynayhi: that is, both his eyes were aching from ophthalmia.

fa-baraʾa: with fatḥa on the bāʾ, pattern faʿala (fa-barāʾa); it may also be read with kasra, pattern faʿila. It means he recovered completely.

aʿṭāhu al-rāyah: he handed the banner to him.

infudh: proceed on your way.

ʿalā rislik: gently, without haste.

bisāhatihim: in their courtyard—the open ground before their forts.

ilā al-islām: namely, to submit to Allah through monotheism, obey Him, and dissociate oneself from polytheism and its people.

wa-akhbirhum … etc.: if they accept Islam, which is monotheism, then inform them of what Allah requires of them thereafter—prayer, zakāh, fasting, pilgrimage, and the like.

li-an yahdiya Allāhu: the clause “that Allah guides” is treated as a verbal noun serving as the subject, whose predicate is “better.”

ḥumru al-naʿam: red camels, the most prized wealth of the Arabs.

Overall meaning of the ḥadīth: The Prophet ﷺ informed the Companions that the Muslims would, on the following day, defeat the Jews at the hands of a man of outstanding virtue who loved Allah and His Messenger and was loved by them. The Companions therefore looked forward to this, each one hoping to be that man.