re: What does firing Comey have to do with stopping the Russian investigation?

What matters is that Trump believed it to be true if the issue is obstruction. If he believed firing Comey would hamper the investigation and that was a motivating factor.

"To build a criminal obstruction of justice case, federal law requires prosecutors to show that a person acted with "corrupt" intent. It does not matter whether the person succeeds in impeding an investigation."

Which it can be argued he likely did.

He told NBC Russia was a motivation for the firing and he told the Russians in that meeting that the firing relieved pressure. Which is why the context of that Oval Office meeting is so important. If it is argued that Trump spoke in a way it was reasonably conceived to be as an order. If the reports that he also asked key people in other intelligence departments to try and push back on the Flynn investigation, and then he fired Comey in an effort that he believed to have relieved some pressure from the investigation, then it's pretty clear cut evidence of obstruction. Is it sufficient to prosecute a case? Not sure. Probably not since sitting presidents can't really be tried criminally. But politically? The bar is broad and low by design anyways.