Chapter 49, Prakriti Khanda of Brahmavaivartara Purana describes birth of Radharani in detail.
Cosmic Radharani, who lives eternally with Sri Krishna in Goloka Vrindavana, was cursed by Sudama to born as human. Sri Krishna was once enjoying Rasamandalam with cowherdess Viraja in secluded place in cosmic Goloka and lakh of Manavatras passed away. Radharani got angry and went to that place. Sudama informed them arrival of Radharani before hand and Viraja escaped from there turing into streams and all oceans in universe are sons of Viraja. Sudama then spoke something which enraged Radhika and cursed him. Sudama also pronounced a counter curse on Her.
न दूट्रा विरजां कृष्ण स्वगृहं च पुनर्ययौ।
जगाम कृष्णस्तां राधां
गोपालैरष्टभिः सह।। २७॥
गोपीभिर्द्वारि युक्ताभिर्वारितोऽपि पुनः
दृष्ट्रा कृष्णं च सा देवी भत्र्सयामास तं तदा॥ २८॥
सुदामा भत्र्सयामास तां तथा कृष्णसंनिधौ।
कुद्धा शशाप सा देवी
सुदामानं सुरेश्वरी। २९॥
गच्छ त्वमासुरीं योनिं गच्छ दूरमतो द्रुतम्।
All the oceans in the universe are the sons of Viraja around.
Thereafter when Radha the great goddess arrived there, she could not
find Viraja or Krsna there. She ultimately returned to her abode.
Thereafter lord Krsna together with eight cowherds reached the abode
of Radhika. Inspite of the gate-keepers of the palace of Radha the
entry of the lord inside the palace was prevented, Krsna forcibly
entered it. At the sight of Krsna, Radhika however denouncing him. At
the same time she found Sudama standing beside Krsna who spoke to
Radha something, which enraged Radha, who pronounced a curse on him,
"At a place far away from here, you will be turned into the form of a
शशाप तां सुदामा च त्वमितो गच्छ भारतम्॥ ३०॥
भव गोपी गोपकन्या
मुख्याभि: स्वाभिरेव च।
तत्र ते कृष्णविच्छेदो भविष्यति शतं समाः॥
तत्र भारावतरणां भगवांश्च करिष्यति।
In his turn Sudama also pronounced a curse on her saying, “You go to
the land of Bharata and turn yourself into a cowherdesses getting
Separated from Krsna for a hundred years. Lord Krsna will incarnate on
earth in order to relieve her burden.”
When both of them felt bad about mutual cursing, Lord Krishna assures them saying details of their births as humans.
कृष्णस्तां बोधयामास विद्यया च कृपानिधिः शीघ्रं संप्राप्स्यसि सुतं मा
Thereafter the merciful lord Krsna cautioned them both with the divine
knowledge and said: "O damsel don't cry, you will get back your son
स चासुरः शङ्खचूडो बभूव तुलसीपतिः।
मच्छूलभिन्नकायेन गोलोकं वै
जगाम सः॥ ३५॥
The same Sudama was reborn as Samkhacuda who happened to be a demon
king and became the husband of Tulasi, who after meeting with this end
at the sight of a trident went back to Goloka.”
Krishna said Radharani would be born without human contact and her parents would be Vrshabhanu and Kalavati.
राधा जगाम वाराहे गोकुल भारतं सती।
वृषभानोश्च वैश्यस्य सा च कन्या
बभूव ह॥ ३६॥
अयोनिसंभवा देवी वायुगर्भा कलावती।
वायुं सा तत्राविर्बभूव ह।। ३७॥
In the Vardha-kalpa, Radhika was born in the village of Gokula in the
family of a Vaisya cowherd. She was born without any human contact.
Her mother during pregnancy carried only the wind. At the time when
the mother was giving birth to the wind, in the meantime Radhika
appeared as her daughter.
अतीते द्वादशाब्दे तु दृष्ट्रा तां नवयौवनाम्।
तत्संबन्धं चकार सः॥ ३८ ।।
छायां संस्थाप्यं तदेहे सान्तर्धानमवाप
बभूव तस्य वैश्यस्य विवाहश्छायया सह॥३९॥
After the expiry of twelve years, finding her becoming youthful, she
was married to a trader named Rayana, at the time of marriage only a
shadow Radha was married to the trader while the real Radha herself
गते चतुर्दशाब्दे तु कंसभीतेश्छलेन च।
कृष्णः शिशुरूपी जगत्पतिः॥४०॥
After the expiry of fourteen years Krsna, the lord of the universe
appeared there as an infant. He was brought to Gokula deceitfully
because of the danger from Kamsa.
कृष्णेन सह राधाया: पुण्ये वृन्दावने वने।
विवाहं कारयामास विधिना जगतां विधिः॥ ४२ ॥
This Rayana was the real brother of Yasodha the mother of Krsna who
happened to be an amsa of Krsna in Goloka and was his maternal uncle
in the sacred forest of Vrndavana. Brahma the creator of the universe
had performed the marriage of Radha with Krsna.
स्वप्ने राधापदाम्भोज नहि पश्यन्ति बल्लवा:।
स्वयं राधा हरेः क्रोडे
Radha whose lotus-like feet were beyond the sight of cowherds even in
dream, the same Radha enjoyed the lap of lord Krsna and her shadow was
very much present in the house of Rāyaņa.
किंचित्काल स वै कृष्ण: पुण्ये वृन्दावने वने।
राधया सह भारते॥४६॥
ततः सुदामशापेन विच्छेदश्च बभूव ह।
भारावतरणं भूमेः कृष्णश्चकार सः॥४७॥
In the sacred forest of Vrndavana in the land of Bharata, Krsna the
lord of the cowherds enjoyed the company of Radha in Vrndavana for a
short duration. Thereafter because of the curse of Sudama both of them
were separated from each other. At that point of time lord Krsna
relieved the earth of her burden.
शताब्दे समतीते तु तीर्थयात्राप्रसङ्गत:।।
ददर्श कृष्णं सा राधा स च
तां च परस्परम्॥४८॥
After the lapse of a hundred years, both Krsna and Radha met each
other while on a pilgrimage.
ततो जगाम गोलोकं राधया सह तत्त्ववित्।
कलावती यशोदा च पर्यगाद्वधया
Thereafter lord Krsna who was well-versed the knowledge of tattva went
back to Goloka together with Radhika and her mother named Kalavati,
besides all the cowherdesses and Yasodha the mother of Krsna.
Previous births of their parents are also described in the same chapter.
देवमाता देवपिता प्रतिकल्पे स्वभावतः॥५४॥
पितृणां मानसी कन्या
वसुदामाऽपि गोलोकाद् वृषभानुः समाययौ॥५५॥
Kaśyapa was reborn as Vasudeva and Aditi reappeared as Devaki in each
and every kalpa, the parents of the god are reborn according to their
nature. Kalavati the mind born daughter of the manes became the mother
of Radha and reaching from Goloka Vasudama to birth in form of
Story of previous birth of mother of Radhika, Kalavati, is mentioned in detail in Chapter 2 Rudra Samhita (Parvati Khanda) of Shiva Mahapurana.
Kalavati was daughter of Kashyapa and Svadha who was daughter of Daksha Prajapati. Svadha had three daughters Mena, Dhyana and Kalavati. Once they went to Sweta Dvipa, place of Lord Vishnu, and didn't stand when Sanatkumaras came there. Sanatkumaras cursed them to born as humans and Parvati Devi was born from Mena, Sita from Dhyana aka Yogini and Radha from Kalavati.
पितृणां तनयास्तिस्रः शृणुत प्रीतमानसाः।
वचनं मम शोकध्नं सुखदं
Sanatkumara said, “O three daughters of the manes, be pleased and
listen to my words which will destroy the grief.
विष्गोरंशस्य शैलस्य हिस्राधारस्य कामिनी॥
ज्येष्ठा भवतु तत्कन्या
You will be born as Parvati, Himalaya, as his eldest daughter.
धन्या प्रिया द्वितीया तु योगिनी जनकस्य च।
महालक्ष्मीनर्माम्ना सीता भविष्यति।।२९।।
The second daughter named Dhanya who is also called the yogini, would
be the wife of the king Janaka. She will, (in due course of from time)
have Sita as her daughter, like Mahālakşmī.
वृषभानस्य वैश्यस्य कनिष्ठा च कलावती।
भविष्यति प्रिया राधा तत्सुता
The youngest one, at the end of Dvapara, shall be the wife of
Vrsabhana- a merchant and daughter will be known as Radha.