The easiest way to think of this may be in terms of some regular relation between time and the size of the universe. Expressing this regular relation as some mathematical formula, it may turn out that, going back from the present in accordance with this formula, we get to a past point of time at which the size of the universe is zero. We would have reason to postulate such a starting point as the origin of the universe if all we know about the universe supports or is at least consistent with our backward extrapolation. Big bang theorists believe that this is (by and large) the case.
The same mathematical formula may be such that, going forward from the present, we get to a future point of time at which the size of the universe is zero once more. We would have reason to postulate such an end point of the universe if all we know about the universe supports or is at least consistent with our forward extrapolation.
There are other conceivable end-of-time scenarios. The amount of stuff in the universe might be declining in accordance with some regular formula which predicts that there will be no stuff left at some future point in time. Or the amount of motion in the universe might be declining in accordance with some regular formula which predicts that all motion will cease at some future point in time. Again, if all we know about the universe supported or were at least consistent with such a forward extrapolation, we might have reason to postulate such an end of time.
To sum up. The passage of time presupposes that something is happening. Something happening in turn presupposes stuff moving in space (or some analogues, on which see Peter Strawson's book Individuals). Time can end by any of these three presuppositions ceasing to hold. We cannot experience such cessation. But we can have reason to postulate it by forward extrapolation -- just as we can have reason to postulate a beginning of time (big bang) by backward extrapolation.
Such extrapolation raises further philosophical issues: What can support our assumption that any mathematical formula (any laws of nature) supported by the evidence we have near the present will continue to hold? And likewise backwards: What supports a big-bang theorist's assumption that the laws of nature she relies on did not evolve but rather held all the way back in time so as to sustain her extrapolation?