The fact that the Republican party was formed by elements of several parties (Northern Democrats, and Whigs and the Free Soil Party) who were most opposed to slavery is probably well know. However the OP/quoted claim also says that:
The Democrats were the pro-slavery party in the North AND the South.
This is pretty debatable in the year 1860 since the Northern and Southern Democrats were bitterly divided that year, so much so that they failed to nominate a common candidate for the presidential race. The two parts of the Democratic party ran different presidential candidates (and not just in the primaries), elected at two different Democratic conventions. A dividing issue was, ahem, how much slavery should be protected by federal law and whether it should be expanded to new states & territories.
The Southern Democrats came to the convention ready to fight for the preservation of slavery—and a candidate other than the Northerner, Douglas. [...] But before the nominations for president and vice president could begin, the convention had to settle on the party platform. When Southern Democrats demanded a plank calling for federal protection of slavery and its extension into new territories, the Northern Democrats refused. They wanted popular sovereignty as the party plank. The Northern Democrats prevailed after two days of intense bargaining, but then the Southern delegates walked out.
In June, the Democrats reconvened in Baltimore, their ranks consisting mostly of Northern states. They nominated Douglas for president and adopted a popular sovereignty platform. Shortly after, the Southern Democrats met in the same city and selected Breckinridge as their presidential candidate.
According to one account, the Democratic Party's doctrine of "popular sovereignty", although a slogan from at least a decade before, proved to be a nebulous-enough concept that its divergent interpretations in the North and South was a large part of the 1860 schism inside the Democratic party. The doctrine roughly held that new states and territories could decide for themselves, without interference from the Federal government, the status of slavery in their lands. The 1860 platform divergence between Southern and Northern Democrats came down to whether territories that were not yet states had the power to reject slave ownership. The Southern Democrats insisted that federal law should protect slave ownership in such not-yet-state territories (thus essentially wedging the door open for slavery's expansion), while Northern Democrats basically refused to legislate along these lines. In further contrast, Republicans rejected the very notion that new states could decide the status of slavery for themselves at all, thus guaranteeing slavery's confinement only to the states that had the instituion already.
At the Charleston convention, the "fire eaters" also insisted on including other adamantly pro-slavery elements in the platform, like the Dred Scott case. According to some accounts, this was done at least in part in order to make the Northerner's Douglas candidacy untenable on such an openly pro-slavery platform. After such proposals were rejected by the majority of the delegates (mostly on a North-South divide), the Southern delegates walked out.
Furthermore, to imply that (in utter contrast) Republicans were uniform in their opposition/solution to slavery around that time is also a bit of an exaggeration. Some quotes (to the contrary) from Lincoln's speeches:
What next? Free them, and make them politically and socially, our equals? My own feelings will not admit of this; and if mine would, we well know that those of the great mass of white people will not.
1858: It is nothing but a miserable perversion of what I have said, to assume that I have declared Missouri, or any other slave State shall emancipate her slaves. I have proposed no such thing.
In the 1860 Republican convention, Lincoln was seen as the more moderate candidate (and won the ticket) because he did not propose any kind of abolition in the Southern states, unlike some of his more radical colleagues.
Lincoln also declaratively supported the Corwin Amendment in his inaugural speech:
I understand a proposed amendment to the Constitution—which amendment, however, I have not seen—has passed Congress, to the effect that the Federal Government shall never interfere with the domestic institutions of the States, including that of persons held to service ... holding such a provision to now be implied constitutional law, I have no objection to its being made express and irrevocable.
By 1865 Lincoln did however propose that former slaves be (even) given the right to vote. As you can see, that's quite a change. So, when someone lionizes Lincoln or the Republicans of 1860, do they mean that slaves should be emancipated only at some indefinite time in the future and that they should probably be never be granted equal status with whites?! It's [so] easy to handwave about history's meanders in order to score cheap political points...