The initial buyers of the shares sold by Anthropic are institutional investors (mostly). That matching process is done by an investment bank they hire.

The trading price you see on the screen is - shortly afterwards the shares become publicly tradable on the exchange (or on all US equities exchanges, for US stocks). So in effect, the price shooting up on the public markets is actually bad for the company in that they might have left money on the table.

It's the price they're going to be offering to the general public to buy ahead of time, through their brokers. They get that price locked in, then once it starts trading it moves as it may. Usually the IPO buyers are not allowed to sell for some time

The price those that buy the IPO get before it starts trading publicly.

E.g. SpaceX is currently trading below its IPO price. If you became a shareholder by buying at IPO and held your have lost money as of today.

The price will settle, but it has to start somewhere.

Well obviously Anthropic as the one offering the shares would care about how much they sell them for.