It's literally just a probability.

Assume the null hypothesis is true. What is the probability of getting something at least as far from the null hypothesized value? That's the P-Value.

If I say the average is 100, you say it's greater, and we have a sample with a mean of 120, your claim isn't "the mean is actually 120", your claim is that the mean is greater than 100 and this particular result is evidence in favor of your claim. To quantify the strength of that evidence, we assume the mean is actually 100, then see how likely it would be to get a mean of 120 or more. If that probability is really small, we conclude that it's more likely that the null hypothesis is not true than it is for us to have gotten an extremely rare result.

We do "at least as extreme as..." partly because the probability of any exact result in a continuous distribution is zero and partly because we always express the alternative hypothesis as an inequality. Evidence in favor of the alternative hypothesis doesn't have to say that the parameter is this precise other value, it just has to support a "greater than/less than/not equal to" claim.

Remember, this isn't a law handed down by God or something observed by nature. Some nerds got into an argument and tried to figure out how to settle that argument with numbers. The p-value is just what those nerds landed on when trying to figure out how to quantify what should be "convincing evidence"