I understand the summary, that we cannot justify expecting the future to resemble the past. What I cannot see is why this is treated as a deep problem rather than an obvious point that we then get on with ignoring.
Science clearly works. What exactly is left unanswered?
Accepted Answer
The force of it is that every attempt to justify induction turns out to use induction.
Ask why the future will resemble the past. The natural answer is that it always has done. But that is an inductive argument, and it is the thing in question. The justification assumes what it is meant to establish.
Hume's point is not that induction is unreliable. He thought we cannot help using it, and that we are right to. His point is that it has no rational foundation, only habit.
"Science works" is an answer to a different question. It is a track record, and appealing to a track record to justify inference from track records is exactly the circle.
Worth separating two things that get run together in most summaries.
The problem of induction is not scepticism about the external world, and Hume is not saying the sun might not rise. He is saying that our confidence that it will has a psychological explanation rather than a logical one.
That matters, because the usual dismissal, "obviously the sun will rise", answers a claim nobody made. Hume would have agreed with you about the sun and disagreed about why you are entitled to say so.
Popper's response was to give up on justifying induction and rebuild science around falsification instead: theories are never confirmed, only not yet refuted. Whether that escapes the problem or renames it is still argued about.
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