remover of the Earth's toil and distress
Earth-goddess Vasudhā, burdened by wicked kings' armies as dense as insects, weeps to Brahmā; removal of her burden (vasudhāyāsa-haraṇa) is the explicit stated purpose of the descent.
वसुधायासहरणो नारदप्रेरणोन्मुखः ।प्रभूष्णुर् नारदोद्गीतो लोकरक्षापरायणः॥
vasudhāyāsaharaṇo nāradapreraṇonmukhaḥ ·prabhūṣṇur nāradodgīto lokarakṣāparāyaṇaḥ
remover of the Earth's toil and distress
Earth-goddess Vasudhā, burdened by wicked kings' armies as dense as insects, weeps to Brahmā; removal of her burden (vasudhāyāsa-haraṇa) is the explicit stated purpose of the descent.
made ready to descend by Nārada's prompting
Nārada visits Kaṃsa in the prison where he holds Devakī and Vasudeva, and tells him that the eighth child will be his destroyer—setting the divine plan in irrevocable motion.
ever-arising, mighty to come forth
At Kṛṣṇa's birth the prison glows without any light source; stars shine with unusual clarity; sacred fires blaze; Vasudeva and Devakī behold the four-armed form blazing with splendour.
extolled in song by Nārada
Nārada loudly proclaims Kṛṣṇa's future glory before Kaṃsa, triggering the chain of events; S.B.G 7 also has Nārada's great instruction to Prahlāda on bhakti toward this same Lord.
wholly devoted to the protection of the worlds
The Milk Ocean prayer explicitly states the purpose: 'O Protect the three worlds'; world-protection (lokarakṣā) is named as the primary reason for each avatāra.