Short answers, pulled from the story.
Babur took over Kabul from the Arghun dynasty in 1504 and re-developed a site that would become Bagh-e Babur. He used it as a guest house for special occasions, especially during summer seasons.
Babur ruled from 1494 until his death in 1530 and founded the Mughal Empire while being buried within the complex. His only child Ruqaiya Sultan Begum lived from 1542 to 1626 and is buried next to him alongside other relatives like his sister Khanzada Begum who lived from 1478 to 1545.
The tomb was destroyed by an earthquake in 1842 after Charles Masson published a description of its fine workmanship in stone carving with high walls and lavish jali-work. The enclosure of white marble curiously and elegantly carved surrounding many monuments commemorating his relatives suffered great liberty taken with the stones employed in the enclosing walls.
Since the 16th of January 2008, the garden has been managed by the independent Bagh-e Babur Trust. Support comes from Kabul Municipality, the Afghan Ministry of Information and Culture, and the Aga Khan Trust for Culture.
About 1,030,000 people visited the site in 2016 while nearly 300,000 people visited the site in 2008 alone. Visitors number up to one million locals and foreign tourists each year at Bagh-e Babur since management changes began.