This isn't true universally, in fact, fair to say it is rarely true [1]. Often the vote is tabulated electronically, but there is always a bipartisan committee overseeing random re-tabulations and recounts as a matter of course. In the process of rolling up results, there are cross-checks with voter rolls that make certain types of ballot stuffing infeasible (that is, an attack would have to be a read/modify/write).
There are also state departments that are tasked with overseeing the general quality of elections that will spot outliers and investigate. It's always possible that corruption or partisan efforts will bypass scrutiny (and there have been several iffy cases of precincts in previous election cycles) but that has been a problem even pre-electronic equipment.
[1] https://ballotpedia.org/Voting_methods_and_equipment_by_stat...
EDIT: the parent post has been modified to be slightly less aggressive in its claims, including limiting the scope to "some states". Also added "And even if they did (as many do), few states have a uniform system of random vote audits", but can easily be refuted [2].
[2] http://www.ncsl.org/research/elections-and-campaigns/post-el...