Suppose candidate X beats candidate Y by 20% according to the machine count. You can then pull a randomly selected subset of the paper ballots and count them. You only need to pick a fairly small subset to establish that the 20% machine count is accurate to within a margin of error much smaller than 20%.
This is called a risk-limiting audit.
If X beats Y by 10% according to the machine count, you need to include more paper ballots in your risk0limiting audit.
In general, the closer X and Y are, the more paper ballots need to be in your audit.
If it is close enough, like it was in Georgia, the risk limiting-audit turns into a full hand recount of all the paper ballots.
Most elections, though, aren't that close, and you only need to hand count a relatively small random sample of paper ballots to with very high probability verify that the correct candidate was declared the winner.
Instead of thinking of them as "voting machines with a paper trail," think of them as "reliable, trustworthy paper ballots that are made clean and uniform using a machine."
The point is that the output is a paper ballot which can be (and often is) hand-counted for auditing and verification purposes. Paper ballots rule.