In order for democracy to work, we have to have 100% confidence in the results of elections - not that we have to like them, but we need to believe in them. And if we don't like them, we can use those results as the basis for improving our losing positions, so they won't be losing next time.
Hillary Clinton of course won the national popular vote. She lost the electoral vote, based on the tallies before the recounts. But that crucial difference, in itself, demands that we be super sure when it comes to the counts in crucial states.
Computer scientists say that the voting patterns reported in those three states are such that there could be some sort of irregularities in the tallies. That doesn't mean that there are - but surely we should investigate further.
It's often said that previous candidates who lost in close Presidential elections didn't go for recounts. Nixon didn't in his close loss to JFK in 1960, and Gore didn't in his loss to Bush in 2000 (when Gore also won the national popular vote).
But Nixon didn't lose to a Donald Trump, and Gore actually did pursue recounts in Florida, which were stopped not by him but by a decision of the U.S. Supreme Court, which Justice Sandra Day O'Connor, who joined the majority decision to stop the recount, later said she regretted.
The election of Trump is the most extraordinarily bad result of a Presidential election in my and I'd bet most Americans' lifetimes (well, certainly a majority). We owe it to ourselves and the future of this country and the world to be sure of its results.