This pattern uses the verb 押す (to push) in its て-form or 切る form to express overcoming resistance. It is metaphorical: you are pushing through opposition as if pushing a heavy door. 押し切って is stronger, implying you cut through and settle the matter decisively, often after a struggle. 押して is slightly weaker and can be used in more continuous contexts, but both mean to proceed despite opposition.

You will most often hear this in discussions about decisions, proposals, or plans where someone faces resistance from others. It is a fairly assertive expression, so it is used when describing someone who went ahead despite objections, not when describing a compromise or a change of mind. In written or formal speech, 押し切って is common; in casual conversation, you might hear 押して or 押し切って as well, but it is still a bit formal for everyday chat.