From Wikipedia :

Halting of stock purchases

On January 28, Robinhood halted purchases of GameStop, AMC Theatres, BlackBerry Limited, Nokia Corporation, and other volatile stocks from its trading platform; customers could no longer open new positions in the stock, although they could still close them.[48] Other brokerages soon followed suit. Many traders were furious, and called for class-action lawsuits in multiple popular Reddit posts.[49] After the markets closed, Robinhood announced it would begin to allow "limited buys" of the affected securities starting the following day, although it was unclear what "limited buys" entailed.[50] Trading platforms such as UK-based Trading212 and Israel-based eToro blocked buys of GameStop and other stock while continuing to allow sales.[51] Webull halted buy orders for stocks affected by the squeeze,[52] but soon thereafter allowed orders to continue.[53] Anthony Denier, the CEO of Webull, stated that increased collateral requirements for their clearing house meant Webull themselves were restricted from opening new positions.[54] Some users alleged that Robinhood was selling shares without consent; Robinhood denied these allegations.[55]

Several brokerage firms, including Robinhood, stated on January 29 that the restrictions were the result of clearing houses raising the required collateral for executing trades.[56][57][58] Because there is a two-day lag between the moment when investors purchase a security and the moment cash and securities are actually exchanged, brokerage firms have to post collateral at clearing houses to guarantee the proper settlement of their clients' orders.[59][57][60] Clearing houses include the Depository Trust & Clearing Corporation (DTCC) for equities and the Options Clearing Corporation (OCC) for options.[56] Clearing houses must have enough collateral on hand to settle a member's outstanding transactions in the event any particular member firm fails—to prevent cascading failures of other members—and can demand additional collateral (i.e., margin calls) from members if market volatility starts to increase.[61][60] Brokerage firms claimed that the increased collateral could not be provided in time, and, as a result, trading had to be halted.[56][57][58] The DTCC, for instance, increased the total industrywide collateral requirements from $26 billion to $33.5 billion,[62] noting that the large trading volumes in specific stocks "generated substantial risk exposures at firms that clear these trades [...] particularly if the clearing member or its clients are predominantly on one side of the market".[62] On January 29, it was reported that Robinhood had raised an additional $1 billion to protect the company from the financial pressure placed by the increased interest in particular stocks and meet the collateral requirements of clearing houses.[63][64]

As of January 29, Robinhood was still imposing limits on the trading of GameStop, AMC, and Blackberry stocks.[62] On January 30, Robinhood announced it had increased the restrictions from the sale of 13 securities to 50, including companies such as Rolls-Royce Holdings and Starbucks Corporation.[65] However, on January 31, Robinhood announced it had removed several of these restrictions and would only limit the sale of eight securities.[66]

It was because fintech companies didn't have enough to cover the trades. When you trade, it's not actually instant. Everything has to go through a clearinghouse. You buy it, the broker puts in a request to buy it, the broker tells you it's been bought, but then it's up to a series of brokers and clearinghouses to make sure the trade actually goes through on the back end. Robinhood, Webull, etc are tech startups, not long standing financial institutions with decades or centuries of capital/clout backing them. They simply didn't have the money or credit to let people keep buying shares.

The SEC investigated this issue afterwards, it wasn't the SEC that told them to do it or stepped in. Yes, regulatory bodies like the SEC are often captured by the industries they're supposed to watch over. But that doesn't mean the SEC is literally on the phone with every broker guiding their hand.

The problem was people was putting a lot on a relatively small and new company. But because reddit and internet it turned into a conspiracy that Robinhood had a grudge against their customers or the CEO had a personal grudge against GME.

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I think I read at the time that it was beginning to turn into not just a short squeeze, but a gamma squeeze. That means the brokerages would have had to participate in buying Gamestop shares in order to hedge against the risks of selling options, which would have pushed the price up even further and the whole thing would have been even more fucked, and could have lead to entire brokerages getting wiped out. So they just said "no" and stopped trading.