The headword is third person singular, present indicative, of the verb eu)/xomai. It is glossed here as in a scholion on Homer, Iliad 1.91 (web address 1, from where it has evidently been quoted); and cf. generally epsilon 3829, epsilon 3830.
In epic, eu)/xomai often means "boast, profess loudly"; in later Greek, it usually means "pray" or "vow" (see LSJ s.v.); cf. epsilon 3828 (and epsilon 3819).