DBZ: why wouldn't Trunks traveling back in time to kill the androids stop his future from being destroyed?

Because Future Trunks comes from a different timeline entirely.

There are two possible theories of how time travel works in science fiction:

1) There is only one timeline, and you travel back through your own timeline.

This means you have the possibility of meeting your past self. However, if you do that but you never met your future self in the past, you would be changing your own history, which could lead to you not existing at all; in other words, you create a paradox.

2) There are multiple timelines, and going back in time will move you to a different timeline. This is the type of time travel that Dragon Ball Z uses. Trunks goes back in time to save Goku from dying from the heart virus so that he could possibly bring Goku to the future to help defeat the Androids.

So basically, we have something like this:

Suppose Future Trunks comes from Timeline A (where Goku dies from the heart virus and the Androids kill everyone). Now let's say the Timeline that we watch in DBZ is called Timeline Z.

Trunks (from Timeline A) goes back in time, ending up in Timeline Z. He planned to save the Goku in Timeline Z and bring him to Timeline A to help defeat the Androids. Of course, after Trunks A spent so much time training with Goku Z and the others to defeat Cell, he didn't need Goku Z's help after all.

The way you're talking about would only work if Trunks A were able to go back in time to save Goku A in the Timeline A. However, if he did that, then the Androids in Timeline A would never have killed everyone, and thus Trunks A would never have needed to go back in time in the first place! This causes a paradox, because if Trunks A never went back in time to save Goku A, then Goku A would die from the heart disease thus requiring Trunks A to go back in time.

So the reason why time travel doesn't work the way you're describing is because it would cause a lot of complicated problems. You can't just go back in time in your own timeline and change the past, because it would also change your present. The only way to avoid a paradox in this case is to create a scenario where the past events can be influenced by your present self, but in a way such that your past self has no idea that it's his future self who's causing those events. (Watch Steins;Gate, for example, since it explores various modes of time travel.)

(Sorry if this is confusing to you, but time travel isn't a concept that can be explained simply.)

Going back in your own time can cause a time paradox that destroys that timeline.

step 1 - Androids kill most of the world's population.

step 2 - Bulma creates time machine to change the past.

step 3 - Trunks goes back in time.

step 4 - Trunks kills androids before they can kill the population.

This is where the infinite time loop begins.

Since the androids were destroyed Bulma had no reason create the time machine. Which means Trunks didn't go into the past which means he didn't destroy the androids which means they did kill people which means we go back to step 2. An infinite loop that they can't escape.

Well...

It's unexplained :p But this is DBZ, so don't think about these things too much.