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: 
There are two logical ways of dealing with time travel paradoxes:

* [**Stable time loop**](http://tvtropes.org/pmwiki/pmwiki.php/Main/StableTimeLoop). When you travel to the past, whatever you do, you won't actually change anything (which pretty much implies there is no free will).

In fiction, the way this is often done is that the time traveller attempts to change the past, but doesn't succeed, because it was him who caused the final outcome in the first place. Or something like that.
* [**Alternate timelines**](http://tvtropes.org/pmwiki/pmwiki.php/Main/AlternateTimeline). When you travel back in time, you cause the creation of an alternate timeline. So there are now two timelines: the one where you left and the one where you arrived. This means that you can do anything and it won't change the original timeline, so no paradoxes are possible.

Though many films and TV shows (including BTTF and Doctor Who) don't use any of the two approaches and instead use approach that doesn't make sense (people slowly vanishing, monsters killing people because change occured, but only sometimes).
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.