Mr. Musk, who made the surprise proposal less than two weeks ago, claimed that Twitter had "tremendous potential" that he would unleash.
He also demanded a slew of adjustments, ranging from loosening content restrictions to eliminating bogus accounts.
The company first turned down Mr. Musk's offer, but it will now put the proposal to a vote among shareholders.
According to Forbes magazine, Mr. Musk is the world's richest person, with a net worth of $273.6 billion, owing mostly to his ownership of electric vehicle producer Tesla, which he also controls. He is also the CEO of SpaceX, an aerospace company.
Mr. Musk said in a statement announcing the transaction, "Free speech is the backbone of a functional democracy, and Twitter is the digital town square where subjects crucial to the future of humanity are debated."
"I also want to make Twitter better than ever by adding new features, opening up the algorithms to boost trust, beating spam bots, and authenticating all people," he continued.
"Twitter has enormous potential, and I'm excited to work with the company and the Twitter community to realize it."
The move comes as politicians and regulators put increasing pressure on Twitter over the content that appears on its site. Its efforts to moderate misinformation on the site have garnered criticism from both the left and the right.
It banned former US President Donald Trump, possibly its most influential user, last year in one of its most high-profile measures, citing the possibility of "incitement of violence."
"A lot of people are going to be really angry with West Coast high tech being the de facto arbiter of free expression," Mr. Musk said at the time.
The right in the United States has reacted positively to the news of the takeover, though Mr. Trump told Fox News on Monday that he had no plans to return to the platform.
"No matter who owns or manages Twitter, the president has long been worried about the power of huge social media platforms," White House spokeswoman Jen Psaki told reporters.
The purchase was described as an "amazing development in the world of social media" by Julian Knight, head of the UK's Digital, Culture, Media and Sport Committee, on Twitter.
"It will be interesting to observe how a privately held Twitter (led by a man who is a free speech absolutist) reacts to worldwide regulatory efforts."
According to BBC environment commentator Roger Harrabin, scientists are concerned about the takeover's possible impact on the climate discussion.
He points out that Twitter declared last week that it would prohibit adverts that dispute the scientific consensus on the climate catastrophe, acknowledging that false information might jeopardize environmental initiatives.
History has a tumultuous past.
Mr. Musk, who has more than 80 million Twitter followers, has a tumultuous history on the network.
US financial regulators accused him of deceiving Tesla investors with his tweets in 2018, allegations that were settled for $40 million and which Mr. Musk continues to reject.
In 2019, he was sued for slander, which he won, after labeling a diver involved in the rescue of schoolboys in Thailand "pedo man" on the platform.
Mr. Musk, who has a history of squabbling with journalists and blocking critics, hinted on Monday that he saw Twitter as a place for dialogue.
Is it possible for Musk to turn Twitter around?
Twitter's shares will be delisted and the company will be taken private as part of the takeover, which is scheduled to close later this year.
Mr. Musk claims that this will allow him to make the changes he wants to the company.
He has proposed, among other things, enabling longer posts and allowing users to amend them after they have been published.
Following the announcement of the acquisition, Twitter's stock rose more than 5% on Monday.
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