A lot of other people have explained guidance technology, but i’m guessing your question is more about the mechanics of actually getting from point a to point b.
Missiles usually have accelerometers installed that tell them how fast they (the missile) are going and in what direction. Their guidance package is used to track the target in relation to their own heading, it also collects information about which direction the target is moving and how fast. After all of that it’s just math to compute and follow an intercept course. And that’s what they do.
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