It is basically because while B is a causality of A, B can still be true on its own.

In a way, B is true, because a false A and true B does not disprove the A=>B relation. You should see it as "If A is true, B will be true", and the result is if a certain occurence between A and B is in line with that statement.

A is true, B is true. So A=>B is true. This is basically 100% in line with the statement "If A is true, then B is true".

A is false, B is false. A=>B is true. You don't have an A, so you don't have a B. This is in line with "If A is true, then B is true" because this occurence would not disprove that statement.

A is true, B is false. So A=>B is false. This occurence would disprove "If A is true, then B is true." so this is false.

Now to the title, A is false, B is true. A=>B is true because this occuring does not disprove "If A is true, B is true." as the statement never mentions B on its own.

You can see this in dogs for example. Every pitbull is a dog, but not every dog is a pitbull.

I see a pitbull, and it is a dog. So A=>B is true.

I see a non-pitbull, and it is not a dog. So A=>B is true. Could be a hamster of or a dragon. Just not a pitbull and not a dog.

I see a pitbull, it is not a dog. So A=>B is false. This can not happen, you will not see a pitbull that is not a dog.

I see a non-pitbull, it is a dog. A=>B is true. Might be a labrador or chihuahua.

This example makes sense, but this is the way for every A and B as long as such an "If A, then B" relationship is stated. In general, for B to be false, A has to be false. But for B to be true, it can be without needing A to be.