Question: There are many movies and TV episodes in which a character "crosses his own timeline", sometimes even meeting his past or future self (e.g. *The Day of the Doctor*). Technically, I should say meeting his past self *and* meeting his future self, I suppose, since this is actually what happens.

When they supposedly alter their own future resulting in their never having travelled in time (as in *Back to The Future* or *Looper*), or narrowly avoid doing so, this is known as the "grandfather paradox".

How are we, the audience, supposed to make sense of this, if we actually think it through logically?

If you perform an action which has an effect that you never survived or were never ever born, then how did you travel in time in the first place? (Although it's not really "in the first place", if you travelled backwards in time...)

I know that there is some quantum mechanics theory relevant to this but the average cinema-goer doesn't know about all that stuff, surely...

Answer: 
My belief is that the audience is guided towards the nonsensical belief that there are two dimensions of time - ordinary time, and "meta-time" - that is, that before "now" in "meta-time", the universe had such-and-such a history, but one second later in "meta-time", history was "changed", and the past became a different past.

This doesn't actually make sense from a physics point of view, and they are usually careful not to be explicit about it.

I am not talking about "personal time" - how time subjectively appears to a character - although it is easy to get confused between "personal time" and this idea of meta-time, because they are usually coterminous from the point of view of a particular protagonist.

The only way to make it really consistent is if the protagonist who "changes the past" is really God, and everything else is effectively just a figment of his imagination, or a dream in other words. Otherwise, as soon as you start thinking about what would happen if someone else also "changed the past", it all falls apart, because the "personal time" theory doesn't fit.
This is a variation of Flater's answer about the universe "preventing" the paradox. At first I was going to comment that it requires a sort of "God" or "Deus Ex Machina" to intervene on behalf of the plot. But think of it slightly differently:

Quantum physics tells us that there is an inherent randomness in the universe. Maybe someone with more knowledge of it can explain it better, but essentially each action and object is actually the average of zillions of probabilities. So while two things might appear identical, at a subatomic level they are completely different.

Now at first glance, the Grandfather paradox is an infinite loop. But what if because of the quantum randomness, each iteration is actually different at a quantum level if not at a macro level. This means that there is an infinitesimal, but non-zero chance that you will fail to kill your grandfather. (Or fail to travel back in time)

But with infinity, even the ridiculously improbable will eventually happen. So even if it takes a gazillion journeys through the time loop, one will eventually occur where you fail to kill your grandfather. And since this one does not prompt a recurrence of the loop, this is the one that will "stick".

You'll forget all the iterations of the loop that have been replaced by this one.
This will only happen with paradoxes of course, because if there isn't a paradox there isn't an infinite time loop.

Because it's a *time* loop, all occurrences of the loop happen at the same time so you don't notice the bazillions of epochs that the loop has been through. So it will appear that the universe or some power does intervene to prevent paradoxes.