Rādhā and Kṛṣṇa are one, and when Kṛṣṇa desires to enjoy pleasure, He manifests Himself as Rādhārāṇī. The spiritual exchange of love between Rādhā and Kṛṣṇa is the actual display of Kṛṣṇa's internal pleasure potency. Although we speak of "when" Kṛṣṇa desires, just when He did desire we cannot say. We only speak in this way because in conditioned life we take it that everything has a beginning; however, in spiritual life everything is absolute, and so there is neither beginning nor end. Yet in order to understand that Rādhā and Kṛṣṇa are one and that They also become divided, the question "When?" automatically comes to mind. When Kṛṣṇa desired to enjoy His pleasure potency, He manifested Himself in the separate form of Rādhārāṇī, and when He wanted to understand Himself through the agency of Rādhā, He united with Rādhārāṇī, and that unification is called Lord Caitanya. This is all explained by Śrīla Kṛṣṇadāsa Kavirāja in the fifth verse of the Caitanya-caritāmṛta.
In the next verse the author further explains why Kṛṣṇa assumed the form of Caitanya Mahāprabhu. Kṛṣṇa desired to know the glory of Rādhā’s love. "Why is She so much in love with Me?" Kṛṣṇa asked. "What is My special qualification that attracts Her so? And what is the actual way in which She loves Me?" It seems strange that Kṛṣṇa, as the Supreme, should be attracted by anyone's love. A man searches after the love of a woman because he is imperfect—he lacks something. The love of a woman, that potency and pleasure, is absent in man, and therefore a man wants a woman. But this is not the case with Kṛṣṇa, who is full in Himself. Thus Kṛṣṇa expressed surprise: "Why am I attracted by Rādhārāṇī? And when Rādhārāṇī feels My love, what is She actually feeling?" To taste the essence of that loving exchange, Kṛṣṇa made His appearance in the same way that the moon appears on the horizon of the sea. Just as the moon was produced by the churning of the sea, by the churning of spiritual loving affairs the moon of Caitanya Mahāprabhu appeared.