Great question! There's a rich history in analytic philosophy grappling with how conditional statements should be understood, categorized, evaluated with respect to truth, etc. The truth conditions you've cited define what is often called the "material conditional".
Most will agree that the material conditional analysis does not match common sense for how conditionals are understood in natural language. You can get a smattering of that by looking at https://en.wikipedia.org/wiki/Material_conditional and https://en.wikipedia.org/wiki/Paradoxes_of_material_implication. I appreciate this longer treatment at http://mit.edu/fintel/fintel-2009-hsk-conditionals.pdf.
The material conditional is widely taught in logic classes in large part because those classes are teaching and working with truth-functional systems and systems of formal derivation. Why that is is worth its own reflection, but with respect to a truth functional framework, there is not a better alternative truth table for the conditional. One way to make it more palatable intuitively is that the systems we're talking about in those classes are often concerned with valid proof, where we're concerned that we have rules that assure us that the truth of the premises guarantee the truth of the conclusion (using truth functional semantics). The material conditional rules fit nicely and intuitively in that context for the formal and mathematical results we'd like to get from those systems, and also intuitively if we somewhat naively state our goals as this: we want only to guarantee that when the antecedent of a conditional is true, it's conclusion is true, and so long as that's not violated, we consider a conditional to be true; it's not violated when the antecedent is false. There's an easy correspondence here between the material truth conditions of a conditional and the validity of derivation: we want a system where, when the premises are true, the conclusion of the derivation is guaranteed to be true.
But outside of that context, how to interpret and evaluate conditionals is a wonderfully large and interesting topic. You might want to look up terms such as strict conditional/entailment, possible worlds semantics, and indicative vs. counterfactual conditionals to get you started looking at more. Or start with that Kai von Fintel link from above.
