मया by me, गुरुसन्निधौ in the presence of my father, स्थिरा firm, प्रतिज्ञा vow, प्रतिज्ञाता is sworn, तदा then, सा that, देवी कैकेयी च Devi Kaikeyi, प्रहृष्यमाणा अभवत् became immensely delighted.

I had taken a firm vow in the presence of my father (about my stay in the forest). And then Devi Kaikeyi was immensely delighted.