Note that logical implication (aka ‘material conditional’) does not translate directly into our intuitive understanding of implication. We often think of implication as synonymous with causality, whereas the material conditional exists irrespective of causality. Mathematics isn’t a science, and its truths are almost always absolute (relative to the chosen axioms), which makes mathematical truth rather than causation its object of study. You can try and shift your thinking in this direction. Try and think of statements like $a\implies b$ as “if $a$ is true, then $b$ also happens to be true” rather than “$a$ causes $b$”.
Note that the main instrument of mathematics is proof, and as used in proofs, the material implication’s purpose is not to formalise some causal connection between mathematical facts, but rather to verify that one of the statements is indeed a fact. Implications of the sort you mention, where the consequent is tautologically true, are of little interest to mathematicians, even though they are correct owing to the truth-functional definition of the material conditional.
To help you come to terms with this definition, remember that the only situation where $a\implies b$ fails to be true is when $a$ is true and $b$ is false. When doing proofs and saying that $b$ follows from $a$, what we really mean is that whenever $a$ is true, $b$ cannot possibly be false (hence is also true). The material conditional guarantees that. And it is that guarantee that is its sole purpose, and what makes making logical deductions in proofs at all possible.