By "the worlds" (al-‘ālamīn), he means the Macrocosm and the Microcosm. As for the Macrocosm, they are the chosen people (ahl al-iṣṭifā’) from the lights of His holiness, and the Microcosm is the spiritual realm.
As for the doubling of the testimony (tathniyat al-shahāda) in the adhān: it is because the first testimony is from the Veil to the Meaning, and the second is the testimony of the creation upon His testimony. This is in the sense of the Almighty’s saying: "God bears witness that there is no deity except Him, and [so do] the angels and those of knowledge, maintaining [creation] in justice" 1 . So "God" here is the Name, and that which occurred through the exception after the negation—by affirming upon Him that there is no deity except Him—is an indication of the Divinity (lāhūtiyya) Meaning (al-Ma‘nā); and by "the angels and those of knowledge," he means the people of affirmation (ahl al-iqrār) from the higher and lower worlds.