It is well established that Tolkien used his fiction as a means of filling gaps and solving riddles in the extant studies of mythology and linguistics.
Most of the time, he chose to do this by supplying a definitive answer to a problem. His work, as a whole, was supposed to represent the lost body of English myth. In a more specific example, he address the problem of elves in northern European myth: these beings are divided into light, dark and swart elves on the one hand and wood and water elves on another. No clear explanation is given of these divisions in surviving myth. Tolkien "solved" the problem with his stories of the various sunderings of the Elves in the Silmarillion.
That example is drawn from an essay on the Silmarillion by Tolkien scholar Tom Shippey. Right after, he makes a throwaway remark that the Silmarils themselves are a similar kind of "solution". He says:
The Silmarils themselves, in my opinion, are an attempt to solve the mysterious riddle of the Sampo, an undefined object often referred to in the Finnish Kalevala - Tolkien was fond of Finnish, modeled aspects of Quenya on it, and furthermore admired the Kalevala as a product of exactly the kind of literary rescue-project he would have liked to see in England.
Academics have apparently been arguing for years about what the Sampo is, and/or represents, and seem to have concluded that a definitive answer is impossible.
The context of this essay - and the bit at the end about the "rescue-project" - would seem to strongly imply that by "solve", here, Shippey means a solid, practical "solution" as per the question of Elvish identity, not an allegorical one.
Other writers have drawn parallels between the Sampo and the Silmarils. And after a cursory read around the subject, it's clear there are significant similarities.
- Both are once-in-a-lifetime objects created by god-like craftsmen at the height of their powers.
- Both are locked away and then stolen, and this action sets in motion a chain of tragic events.
- Both end up being lost in the sea.
There are other examples, with these being the strongest.
However, while all this strongly suggests that Tolkien's story of the Silmarils was inspired by the Sampo, it does not seem clear how it "solves" the mythical problem of what the Sampo is and what it represents. So, what does Shippey mean by his remark that the Silmarils address this problem?