You believe bread will nourish you tomorrow because it always has. You believe the sun will rise, that dropped objects will fall, that fire will burn. Every one of those beliefs reaches beyond what you have observed to what you have not — and Hume's question is what entitles you to make that step.
Notice first that it is not a matter of logic. From "every observed emerald was green" it does not follow that the next will be; there is no contradiction in supposing otherwise, in the way there would be in supposing a bachelor married. The inference is not deductively valid, so if it is justified at all, it must be justified some other way.
The natural reply is that nature is uniform — that the future generally resembles the past, and regularities persist. That would license the inference. But how is that established? Not by logic, since a chaotic universe is perfectly conceivable. So presumably by experience: nature has been uniform so far. And there the trap closes. Using past uniformity to establish that the future will resemble the past assumes exactly the principle in question. The justification is circular.
Hume's own conclusion was psychological rather than sceptical in the practical sense. We do not infer these things by reason at all, he argued, but by custom or habit: repeated experience produces an expectation, and that habit is what actually drives our reasoning about matters of fact. He did not think anyone could or should stop reasoning inductively — he thought nature had wisely not left something so essential to our shaky powers of argument.
The responses form a map of subsequent epistemology. Popper accepted the argument fully and concluded that science does not use induction: it makes conjectures and refutes them, so no inductive justification is needed. Critics reply that choosing to rely on a corroborated theory rather than a refuted one seems to reintroduce induction through the back door. Pragmatic responses, notably Reichenbach's, argue that if any method works induction will, so we lose nothing by using it. Bayesian approaches formalise how evidence should update degrees of belief, though they still require prior probabilities from somewhere. And Goodman made matters harder with his "grue" puzzle, showing that the evidence supporting "all emeralds are green" equally supports a bizarre alternative predicate — so the problem is not merely justifying induction but explaining which inductions are the right ones.
What makes this the most respectable unsolved problem in philosophy is its combination of simplicity and reach. Nobody proposes acting on it. But it sits underneath every empirical claim ever made, and nearly three hundred years of effort have not produced a solution that its proponents can defend without eventually leaning on the very thing in question.