This Hindupedia information page states that Garuda once lost his wings due to a curse by Swayamprabha, the female ascetic that Hanuman and other Vanars met while looking for Sita. Can anyone shed light on why did this lady curse Garuda to begin with and what prayers did he perform to get them back?
The story of the loss of Garuda's wings is narrated in Chapter 81 of the Nagara Khanda of Skanda Purana.
Once Garuda visited Lord Vishnu with a marriage alliance to help a Brahmana friend of his. [This is kind of a story within a story so I'm going to omit some details]
That time he notices an old ascetic woman sitting near Vishnu. He asks about her. Vishnu tells him the woman is Sandili (Hindupedia says Swayamprabha was Sandilya's daughter so this must be the same woman) a venerated ascetic lady. And Scriptural reference to Swayamprabha being Shaandili is this-
Mahabharata Bheeshma Parva Chapter 8
सर्वरत्नमयं चैकं भवनैरुपशोभितम् ।
तत्र स्वयंप्रभा देवी नित्यं वसति शाण्डिली ॥ verse 9 ॥
Sanjaya tells DhritaRaastra-
In that one mountain, which is full of all gems and is beautified with Palaces,
there Devi SwayamPrabhaa who is Shaandili (id est Daughter of Shandilya), always dwells.
Here is story how Garuda was cursed by SwayamPrabhaa.
26.At the close of the narration, Garuda looked at the aged lady endowed with refulgence and spoke to Purusottama:
27.“O Jamārdana, O most excellent one among Suras, this elderly lady sitting near you seems to be someone not seen before. Do tell me who she is. Where has she come from? Why has she come?”
Śrī Bhagavān said:
28-29. O most excellent one among birds, this is the elderly maiden well-known in this world by the name Šāņdilī. She is omniscient and strictly adhering to celibacy. She is endowed with the power of penance. She is venerated by all the Devas. O leading bird, there is no woman like her in all the three worlds
Garuda seems incredulous at the prospect of a woman being a celibate ascetic and makes various assumptions about womenfolk. His speech angers Sandili and she with her ascetic power causes Garuda's wings to become deformed and then he cannot move.
Suta said: 30. On hearing his words, O Brahmanas, the Lord of the birds looked at her for a long time, then laughed and ultimately spoke to Vasudeva:
31-32. It is not unusual that a very great penance is performed. Similarly if gifts are made it is not a wonder. Further if a battle is fought by able warriors, what is surprising in that? But, O Purusottama, this is unusual that one maintains celibacy especially during the prime of youth.
33-35. More so in the case of women. I do not have any faith in this respect. Emotional outlet is essential in the case of lower animals. Youth is not for suppression of rights. If women do not get access to men anywhere, they will indulge in mutual sexual alliance when afflicted by the arrows of Kama. In the absence of suitable men, they will consider a leper, a sick one, and an old man and one short of a limb or two on a par with the Five arrowed One (Kāma).
37.If women keep a limit, it is not due to the fear of the other world. Except the fear of the king or that of the elders (there is nothing to restrain them).
38.On hearing these words of his, Sandili, the celibate woman, had in her heart the surging emotion of anger, although she was under the vow of silence.
39.In the meantime both the wings of the Lord of the birds got destroyed and he became mutilated in shape.
40.Though he was really ferocious and devoid of all ailments, he could not go ahead even a step. He was no better than a mass of flesh.
Vishnu asks Sandili why she did it and also asks for a means of atonement. Sandili directs Garuda to worship Lord Shiva. After installing a Shivalinga and worshiping the Lord, Garuda regains his wings, stronger than before.
Śrī Bhagavān said:
6.O highly esteemed one, only a general statement was made by him about all the women. So, why was he made thus by you?
7.O Janardana, he looked at my face and smiled. O preceptor of the universe, a censure of woman-folk was also made by him wantonly.
8.It is for this reason that he has been thus chastised by me. O Kesava, it is not mental, verbal or physical attack.
Šī Bhagavān said:
9.Still, O sinless one, O auspicious lady, if you honour me, do a favour unto this fellow at my instance.
10.O Lord, whatever is conceived by me in my mind, good or otherwise cannot become otherwise, especially if my anger is aroused.
11.Hence at my behest, let him propitiate Sankara for the restoration of the wings. No one else has enough competence vested in him to grant him (the wings).
12.Otherwise, O Lotus-eyed One, he will have to go permanently like this in the world. I am telling you the truth.
17.On hearing it, Garuda immediately adopted Pāśupata Vrata after installing Lord Isana and then propitiated him. He thereby caused delight to him.
21.Thus he continued performing the Vrata rites, Japa and Puja. Then at the end of a thousand years Mahesvara became pleased. He said, "O most excellent bird, I shall grant boons. Choose whatever is desired."
22-23. See my plight, O Isana, brought about by Sandili. My wings have been caused to fall down. This is what I request you further, O Hara, that you should stay here always in the Linga at my instance, unhesitatingly, if you are to grant me what is desired by me.
Śrī Bhagavān said:
24-25. From today onwards my stay shall be here in this Linga. You will also be endowed with that original form, particularly possessing special strength and speed. This shall undoubtedly take place with my favour, O bird. After saying this to him, the Lord himself touched him with his hand.
> 26.At the very instant beautiful wings grew. So also divine hairs comparable to gold appeared thereon.
27.Then the delighted bird bowed down to the Lord. Afterwards he took leave of Mahesvara and went to his abode.
34.Hence particularly in Kali age, the Lord named Suparņa (Suparnakhya) should be visited by persons endowed with faith with all efforts.
The deity is known as Suparneshwar.
A similar story is depicted in Chapter 113 of Udyoga Parva of Mahabharata.
Here Garuda doesn't have to worship Shiva and even the reasons why he lost his wings are different. It is Sandili herself who grants his wings back to him.
"Narada said, 'Alighting then on the peak of the Rishabha, the Brahmana and the Bird beheld a Brahmana lady of the name of Sandili, engaged there on ascetic penances. And Galava and Garuda both saluted her by bending their heads, and worshipped her. [...]
And Garuda, from desire of leaving that place, upon awakening, found that his wings, had fallen off. Indeed, he had become like a ball of flesh, with only his head and legs. And beholding him come to that plight, Galava sorrowfully enquired, saying, 'What is this condition that has overtaken thee as the consequence of thy sojourn here? Alas, how long shall we have to reside here? Hadst thou harboured any evil and sinful thought in thy mind? It cannot, I am sure, be any trivial sin of which thou hast been guilty.'
Thus addressed, Garuda replied unto the Brahmana, saying, 'Indeed, O regenerate one, I entertained the thought of carrying away this lady crowned with ascetic success from this spot to where the Creator himself, the divine Mahadeva, the eternal Vishnu, and both Virtue and Sacrifice personified, live together, for as I thought this lady should live there. I shall now, from desire of doing myself good, prostrate myself before this holy lady, and pray unto her, saying,--with a heart full of pity, I had, indeed, entertained such a thought. Whether I acted rightly or wrongly, even this was the wish, evidently against thy own, that was cherished by me from my respect for thee. It behoveth thee, therefore, to grant me forgiveness, from the nobility of thy heart.'
That lady became gratified with that prince of birds and that bull of Brahmanas. And addressing Garuda, she said, 'Fear not, O thou of beautiful feathers. Resume thy wings, and cast off thy fears. I was contempted by thee, but know that I do not pardon contempt. That sinful being who entertains contempt for me, would speedily fall away from all blissful regions. Without a single inauspicious indication about me, and perfectly blameless as I am, I have, in consequence of the purity of my conduct, attained to high ascetic success. [...]
Go thou, O blessed prince of birds, whithersoever thou wishest, from this place. Never entertain contempt for me, and take care that thou dost not contempt women that may even be truly blamable. Thou shalt again be, as before, invested with both strength and energy.'
At these words of that lady Garuda had his wings again, and they became even stronger than before. And then with Sandili's leave, Garuda with Galava on his back took his departure.
[Note: Galava is a friend of Garuda, who was in search of 1000 horses with white body but one black ear. This was asked by Vishwamitra, who was Galava's guru.]