They say

In recent decades, there have been over 1,200 known instances of voter fraud due to which 20 US elections had to be overturned to declare a new winner

with a link to a web page on the Heritage Foundation's website. I picked ten states at random and looked at the most recent two cases on each. I've boldfaced every case where it seems plausible that more than 10 votes were affected in any election:

Washington. Two individuals each of whom attempted to cast a small number of fraudulent votes. (Can't tell the affiliation of the first; second tried to vote for Obama despite not being registered.)
North Dakota. Two individuals each of whom attempted to cast a small number of fraudulent votes. (Can't tell the affiliation of either; both cases look as if they may have been inadvertent rather than deliberate fraud.)
Nevada. Renaldo Johnson submitted six fake names on a petition aiming to get the Green Party's nominee on the presidential ballot. It's not 100% clear to me exactly what Tina Marie Parks did (it seems to have been multiple different things, not all clearly described in that article) but it looks like the main thing was a small number of fake voter registrations.
Pennsylvania. It's not clear to me how much Harry Maxwell did (the story describes one fraudulent ballot; his confession seems to describe either trying to do it repeatedly or actually doing it repeatedly and I can't tell which) nor his party affiliation. Calvin Mattox and his pals ran a more substantial attempt to screw up voting in favour of Democratic candidates (there was some weird situation where the Democratic incumbents were thrown out because of some sort of misdeeds, no Democrats at all were on the ballot, and a write-in Democratic candidate eventually won).
Texas. Charles Jackson gave false information on a registration form. Armando O'Caña was accused of bribing voters and other irregularities in order to get elected mayor of the city of Mission. The election was overturned and then on-overturned on appeal. Party affiliation not 100% clear in either case but O'Caña was represented by the chairman of the Texas Democratic Party in his appeal, so he's probably a Democrat.
Florida. Bret Warren stole some absentee ballots, filled them in himself, and submitted them. Mia Antoinette Howells and others seem to have done a wide variety of bad things in a mayoral electon in Eatonville; they were found guilty on multiple charges. Curiously, that article says they were charged with 22 third-degree felonies and their actions may have affected "at least five voters", which seems rather few. (But then the whole population of Eatonville is only about 2k.) No idea of Warren's affiliation, but Howells et al were Democrats.
Illinois. First two entries relate to the same case: a family tried to register to vote at a vacant address. Not obvious what their affiliation is.
Maine. Derek Abbott lives on a border and voted twice in several elections. Delmer Terrill voted twice in one election.
Tennessee. First two entries relate to the same case: Brian Hodge and Betty Best tried to buy votes for a Republican candidate for sheriff of Monroe County. The number of votes affected seems to have been 13.
California. Norman Hall tried to buy petition signatures from homeless people ("spanning interests of several political parties", and it seems like he was being paid but so far as I know it isn't known by whom). April Atilano submitted fake registrations for a gubernatorial election. The article's a little ambiguous but it looks like she was doing it for the Democrats.

None of these is on a large scale. (Largest is probably Mattox et al in Pennsylvania.) Few if any are in general elections. Most of them are isolated cases of an individual or family trying to vote multiple times or register somewhere where they're not entitled to vote. The average size of fraud in the cases I looked at is certainly no bigger than 10; if they are representative in this respect of the 1200 cases the Heritage Foundation alleges, this amounts to maybe 12k fraudulent votes over the last ~20 years, mostly in local elections.

One fraudulent vote is one too many. But my conclusion from the Heritage Foundation's data is that on the whole this is not something we need to be worrying about more than we already are. It's rare, it's usually small, it seldom affects election results, the ones it does affect are generally pretty unimportant elections. There's certainly nothing here that makes it more credible that anyone's fiddling general election results by multiple thousands of votes. (That doesn't mean no one is. But if they are, it's probably quite a different phenomenon from the small-scale cheating the Heritage Foundation is reporting on.)

It looks as if election fraud in the Heritage Foundation's data leans Democratic. I don't know whether this is (1) small-n noise, (2) the Democrats doing more election fraud, (3) the Democrats being worse at not getting caught, (4) the Republicans chasing it harder and targeting Democrats, (5) the (right-wing) Heritage Foundation looking harder for Democratic cases, or (6) something else. In any case, it seems like a leaning rather than one party doing almost all of it, so that estimate of 12k fraudulent votes over the last 20 years will have fraud in all directions.

For the avoidance of doubt: I am not commenting on that article as a whole, just one little bit that caught my eye.