The purpose of a paradox, if it has one, is to display the limitations of our habits of thought.
Given our normal habits of thought, the question: "Assuming time travel is possible, what happens if someone goes back in time and kills their grandfather?" appears difficult. We are tempted to ask ourselves what could stop them doing it, and find nothing.
A similar question "Assuming carpentry is possible, why should someone not make a round square table?" appears less difficult: there is no need for anything or anyone to stop round squares being made, because if something is round then it is not square, and vice versa.
So let us rephrase the time-traveller question to "Assuming time travel is possible, why should someone not be both alive and dead on the date of conception of a time-traveller's parent?" This is the time-traveller's intention, since his grandfather was alive on that date and he desires him also to have been dead. The answer is the same as that to the question concerning the carpentry of round squares: there are no contradictions. Carpentry lets you shape wood, but within limits, including those of logic. Time travel lets you shape the past, but within limits, including those of logic.
The habit that leads us into difficulty is our natural tendency to be impressed by stories: normally, knowing how something happens is useful for understanding it. The purposeful activity involved in climbing into a time machine and engaging in targeted assassination appears to make the desired outcome comprehensible. But this is only an appearance: the desired outcome is still a contradiction.
Here is a similar story: there once was a joiner who loved to make round tables. In fact, he only ever made round tables. But it happened that he was very poor, and a rich man offered him a lot of money to make a square table. So the joiner thought things over, and had a wonderful idea: he made a beautiful round table for himself, and also made it square for his customer. The customer was very happy, and paid him twice, because, as he said, he had both a round and a square table.
They say that the joiner had a grandson who built a time machine...