Why did Rishi Markandeya recite the story of Lord Rama to Yudhishthira?

 After Jayadratha made an attempt to forcefully kidnap Draupadi, Yudhishtira fell into grief. He lamented about the sufferings Draupadi had to face. Then he asked Markandeya if there was anyone more unfortunate than him.

Yudhishtira's question

This life in the forest is full of misery. We sustain ourselves through hunting. This means that those who live in the forest cause violence to those of the deer family. This exile has been brought about by relatives who resorted to falsehood. Is there any other man who is more unfortunate than I? Have you seen, or heard of, any such one earlier?”’

• 554 (257), Draupadi Harana parva, Mahabharata.

Markandeya told Yudhishtira that Rama of Ikshwaku dynasty faced a great misery

Markandeya said, “O bull among the Bharata lineage! Rama confronted incomparable misery. His wife, Janaki, was forcibly abducted by a rakshasa. Having swiftly killed the vulture Jatayu by resorting to maya, Ravana, Indra of the rakshasas, abducted her from the hermitage, travelling through the sky. Resorting to the strength of Sugriva, Rama bound a bridge over the ocean, burnt Lanka with his sharp arrows, and obtained her back.”

• 555 (258), Draupadi Harana parva, Mahabharata.

Following this, Markandeya narrated the entire life story of Sri Rama to Yudhishtira, how he was exiled, how his wife was abducted by Ravana and how Rama went to Lanka and destroyed the Rakshasas.

After this, Markandeya comforted Yudhishtira by reminding him that Rama overcame his enemies with the help of Vanaras. Yudhishtira has his powerful brothers to always aid him.

Who cannot be overcome in battle by someone who has Dhananjaya as his brother? Bhima is terrible in his valour and is foremost among strong ones. Madri’s two twin sons are young and great archers. O scorcher of enemies! With such aides, why do you grieve? With such aides, you can vanquish the soldiers of the wielder of the vajra, together with the Maruts. O bull among the Bharata lineage! With such great archers as aides, who are the equals of the gods, you will vanquish all your enemies in battle. Look at Droupadi Krishna. She was abducted forcibly by the evil-souled Saindhava, who was intoxicated with his valour. But these great-souled ones accomplished the difficult task of obtaining her back, after vanquishing and subjugating King Jayadratha. Rama got Vaidehi back without any such allies. Through his terrible valour, he killed the rakshasa Dashagriva in battle. O king! Use your own intelligence to think about this. Monkeys and black-faced bears were his allies, creatures from a different species. O best of the Kurus! O bull among the Bharata lineage! Therefore, do not sorrow. O scorcher of enemies! Great-souled ones like you do not sorrow.”

• 573 ( 276), Draupadi Harana parva, Mahabharata.

Thus through the story of Rama, Markandeya freed Yudhishtira from his sorrows.

What was the reason for the downfall of the Kuru dynasty after Yudhishthira?

 Kuru kingdom flourished during the time of Parikshit and his son Janamejaya.

After a few generations, their power was shattered by the Invasion of the Shalva tribe

The Kuru realm survived under Pariksit’s descendant, Janamejaya Pariksita, Janamejaya’s sons, his grandson Augrasainya, and probably beyond this. A closer reading of the texts yields more results for this still very hazy picture of Vedic history: the Kurus were overcome by the (probably non-Vedic) Salvas who “dispersed the Kurus from Kuruksetra” (JB 2.206, SSS 15.16.11 12,149) — a fact completely overlooked by the historians of old India.

• Page 21, Early Sanskritization: Origin and Development of the Kuru state.

The Puranas mention about furthur calamities like a flood that destroyed Hastinapura

His son will be Sahasranika whose son will be Asvamedhaja; his son will be Asima-Krsna, whose son will be Nemicakra. When Hastinapura will be washed away by the floods of the river (Ganga), he will settle down at Kausambi (Kosam, near Allahabad). He will have Citraratha as his son, of whom a son Kaviratha will be born.

• Chapter 22, 9th Skandha, Srimad Bhagavatam.

Chandogya Upanishad describes furthur destruction that befell the Kurus

At a time when the Kuru country had been devastated by hail (alternatively locust or thunderstorms), Usasti Cakrayana, with his young wife Atiki, lived in a deplorable condition in the village of the wealthy landowners.

• Chandogya Upanishad 1.10.1

The Bharata royal family was gone by the time of Brihadaranyaka Upanishad as sage Bhujya asks what happened to the Kurus and Yajnavalkya replied that they went to heaven

When we asked him about the limits of the world, we said to him, “Where were the descendants of Pariksit?” And I ask you, Yajnavalkya, where were the descendants of Pariksit? (Tell me) where were the descendants of Pariksit?’

Yajnavalkya said, ‘The Gandharva evidently told you that they went where the performers of the horse sacrifice go.’

• Section 3, chapter 3, Brihadaranyaka Upanishad of Shukla Yajurveda.

The Kuru kingdom existed in a weakened state even during the time of Buddha. However political power shifted first to Panchalas, then to Kosala and finally to Magadha. Whatever royal family remained there were probably uprooted by Mahapadma Nanda.

The Kurus still existed in Mauryan period, but as a minor republic (corporations or Sangha)

The corporations of the Licchivika, Vrjika, Mallaka, Madraka, Kukura, Kuru, Pancala, and others live by the title of "Raja".

• 11.1.5, Arthashastra.