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.
If the time travel is branching, then it needs no explanation, but if it's linear, then I'd imagine it's because time travelling protected them? For example, and I hope this is not just in the CW show, the Reverse Flash, mortal enemy of the Flash (I hope you could've figured that one out on your own), is able to basically exist in every time period because the Speed Force protects him and there's versions of him... everywhen?

But Marty began to disappear in BTTF, whereas Reverse Flash isn't affected at all in his time travelling.

Different worlds have different rules so it depends on the specific universe.