Robinhood doesn't actually execute any trades. They sell user's trade orders to Citadel, a trade executor. Citadel then buys the shares on the market, and sells them to Robinhood for a slight markup. It's how Robinhood offers trades for $0 fees. You pay pennies more per share, but don't have to spend $5 per trade.
Citadel, and other trade executors, are refusing to buy shares for retail traders. Coincidentally, Citadel also bailed out Melvin fund for their short position in GME. So, Citadel has an interest in not letting the price go up any further. And citadel controls trade execution for dozens of firms.
This is definitely illegal. But Citadel is betting that the resulting SEC fines from this illegal manipulation will be less than the loss they would get if they didn't suppress the price.