Nothing... at the moment.
According to the item schema, mystery solving is an actual attribute, but it would appear to have no effect in game. Obviously, it's not clear at this point, but a future update might actually make the attribute do something.
"name": "mystery solving time decrease",
"description_string": "Reduces mystery solving time by up to %s1%",
No item sets since Australian Christmas have had effects that change the attributes of weaponry/damage resistance, so it would be odd for Valve to release a new item set with beneficial stats, suggesting that it may just be a joke attribute, and part of the theme of the item set.
As the TF2 wiki also notes:
The item is a reference to Sir Arthur Conan Doyle's fictional detective Sherlock Holmes. Specifically, it alludes to the phrase "three pipe problem" introduced in the short story The Red-Headed League, and refers to the amount of tobacco needed to ponder over and deduce the solution to a particular puzzle.
which further suggests it could just be part of the item set theme.