The context is the Rājasūya Yajña. Reproducing the Sanskrit verses with their hindi translations here:
गवां शतसहस्त्राणि शयनानां च भारत ।
रुक्मस्य योषितां चैव धर्मराजः पृथग् ददौ ॥५२॥
प्रावर्ततैवं यज्ञः स पाण्डवस्य महात्मनः ।
पृथिव्यामेकवीरस्य शक्रस्येव त्रिविष्टपे ॥ ५३॥
Hindi Translation:
भारत ! धर्मराज युधिष्ठिरने एक लाख गौएँ, उतनी ही शय्याएँ, एक लाख स्वर्णमुद्राएँ
तथा उतनी ही अविवाहित युवतियाँ पृथक्-पृथक् ब्राह्मणोंको दान की ।। ५२ ॥
इस प्रकार स्वर्ग में इन्द्रकी भाँति भूमण्डलमें अद्वितीय वीर महात्मा
पाण्डुनन्दन युधिष्ठिरका वह यज्ञ प्रारम्भ हुआ ॥ ५३॥
A Rājasūya Yajña is one of foremost of the Yajñas - which forms the foundation of the Śrauta practises and rituals of the Vedas. A Yajña (and for that matter any "sacred" act of Vedic ritual), that too, to be performed by a King necessarily enjoins the concept of dānam and other charitable works.
This is what Śrīmada Bhagavad Gītā says on the importance of Yajñas and dānam :
यज्ञदानतपःकर्म न त्याज्यं कार्यमेव तत्।
यज्ञो दानं तपश्चैव पावनानि मनीषिणाम्।।18.5।।
English Translation By Swami Gambirananda
18.5 The practice of sacrifice, charity and austerity is not to be abandoned; it is surely to be undertaken. Sacrifice, charity and austerity are verily the purifiers of the wise.
The alms given to Brāhmiṇas, which are described in those Mahābhārat verses, viz. Cows, Beds, Gold Coins & young unmarried damsels { i.e., Virgin Girls ( kanyā - कन्या )} – each of these four form the part of the Mahādānam - the great gifts, which besides these four contain 6 or 12 more other items in the Mahādānam itenary list amounting to a list of either 10 or 16. Some scriptures like the Skanda Purāṇa mention upto 16 individuals items which can be gifted to qualify as Mahādānam while other texts like the Agni Purāṇa give a list of 10 items which qualify as Mahādānam. Other Purāṇas too might have similar lists.
I'm quoting the 16 item list from the Skanda Purāṇa.
Chapter 208 - Brāhmaṇas Eligible for Dāna:
[Section 1 - Prabhāsa-kṣetra-māhātmya, Skanda Purāṇa]
गावः सुवर्णं रजतं रत्नानि च सरस्वती ॥
तिलाः कन्या गजोश्वश्च शय्या वस्त्रं तथा मही ॥ ११ ॥
धान्यं पयश्च च्छत्रं च गृहं चोपस्करान्वितम्॥
एतान्येव महादेवि महादानानि षोडश ॥ १२ ॥
11-12. O great goddess, these are the sixteen Mahādānas: Cows, gold, silver, jewels, Sarasvatī (i.e. learning), gingelly seeds, a virgin, an elephant, a horse, a bed, a garment, earth (a plot of land), grain, milk, an umbrella, and a house fully furnished.
The Dharma Śhāstras are very clear and strict about the age and virginity of the "bride-to-be" , i.e., she must be a virgin and younger than the groom. And thus, the importance of specifically giving "young girls" as brides to the eligible Brāhmiṇas , and not "old-women", as part of dāna in the great Rājasūya Yajña.
Also, kindly note that the interpretation - "that a thousand damsels were given to each Brāhmiṇa", is wrongly interpreted from the English translation.
This wrong interpretation is a very common thing with most English translation. Sanskrit verses need to be clearly analyzed, in these kinds of cases.
As I have linked already in the beginning itself, the original Sanskrit verses and their hindi translations too.
That makes it perfectly clear that -
Each Brāhmiṇa thus would have been gifted a cow, a few gold coins, a bed and a maiden as wife (not thousands). Each of the four entities were "gifted" in singular quantities to each of the thousand Brāhmiṇas. A one-on-one "injective-function" mapping exists for each gifts to each of the Brāhmiṇas.
Thus, tl;dr
The giving of “young damsels” is to be interpreted as Kanyādāna.
The girl is given as a dharmically wedded wife, as a virtuous wife ( sahadharmacāriṇī ) to an eligible bachelor ( brahmacāri ) Brāhmiṇa.