Let's try an analogy:

We want to know if the null hypothesis "Noetsi was driving a car at 3pm on Thanksgiving Day" is true.

The alternative hypothesis is that Noetsi was not driving a car at 3pm on Thanksgiving Day.

We're given the piece of information that Noetsi was in a car at 3pm on Thanksgiving Day.

The probability that Noetsi was in a car at 3pm on Thanksgiving Day, given that the null hypothesis "Noetsi was driving a car at 3pm on Thanksgiving Day" is true, is equal to 1. (This is sort of like a p value; we could also call it a conditional probability).

However, the probability that Noetsi was in a car at 3pm on Thanksgiving day, given that null hypothesis isfalse, isnotzero.

It's quite possible that Noetsi could have been riding as a passenger or parked in a car, but not driving. Therefore, even though the "p value" is 1, we can not be certain that the null hypothesis is true in this case.

Does this make sense?

For a more conventional example:

Imagine that we're interested in the correlation between variables A & B in population Z.

Our null hypothesis is that the population correlation is zero.

For tractability's sake imagine we set a specific "point" alternative hypothesis: that the population correlation is 0.3.

We then randomly sample 25 individuals from population Z, and (remarkably) find that the sample correlation is 0.

The probability of observing a sample correlation this or more different from zero under the null hypothesis (i.e. conventional p value) is of course 1.

We thus might be tempted to conclude that the null is definitely false.

However, the probability of observing a sample correlation this different from 0.3, given that the alternative hypothesis that the population correlation is actually 0.3 is true, is not zero. It's in fact not all that unlikely: the probability is around 0.15. So while the evidence favours the null hypothesis, we can by no means be certain that the null is true.