Twitter had traded above that price as recently as five months earlier, and had traded in the $60s and even $70s for most of 2021. Twitter's stock had cratered in early 2022 along with many other tech stocks (Facebook went from $330 to $220 in that period).
At the time, the Board could have argued that the current stock price was artificially low due to market forces, and that it wasn't an accurate reflection of the company's prospects. That's especially true because Boards have access to non-public information and longer term forecasts.
When the board agreed to the merger, it signaled to investors that sophisticated insiders on the Board and their financial advisors did not believe the stock would be reasonably likely to reach the offer price in the near/medium term, on a risk adjusted basis, and thus the acquisition is superior. That's important intel for investors, and one of the reasons why failed mergers are so risky.
The 38% is just the one day premium. The offer was at a discount to the stock price less than six months earlier. It was possible that the stock price had overreacted to a tech sell off, and if the market had rebounded by the summer the bid could have looked genius.
Market pricing in M&A is somewhat opaque and different than trading prices because it's based partly on synergy value to a bidder. To use your house example, if the house next door was willing to pay $1.5 million for your house, then the $1.38M is not above market even if Zillow and the list price say it's $1 million.
There were other potential suitors for Twitter, like Microsoft, Disney, Salesforce and private equity. They might have valued it more than Musk based on the benefits it brought their firms. Twitter would be a lot more valuable inside Amazon, Google or Meta but there were probably antitrust issues there.
Plus Twitter's board and management team had access to a lot of material non public information that investors did not. Even if you thought the market price in early April accurately reflected all the public info on Twitter, the Board was looking at 5 year projections and much more detailed pricing/demand/customer info.
The Delaware Courts give a lot of deference to board judgment in these situations, and Twitter's board would have been well within their rights to say "no, that 38% is not good enough" if that was their conclusion after a good faith and reasonable process.
So he just made it private?
It's a rather dramatic situation.