ēnam Rsyasringa, anta:puraṅ pravēśya having taken him to the inner apartment, śāstrata: traditionally as enjoined by the scriptures, pūjāṅ welcome, kṛtvā having made, tadā then, tasya upavāhanāt for bringing him here, ātmānam himself, kṛtakṛtyam has accomplished his object, mēnē thought.
He (king Dasaratha) took him in to the inner apartment and extended welcome. He thought that he has as enjoined by the sastras accomplished his objective by bringing him (Rsyasringa).