There are other floristic disjunctions in the world, but this one was the first described and studied. As long ago as the 1750's botanists were noticing the commonality of plants in the two regions, and by the 1850's had enough data and collected enough specimens that they were able to conclude that the flora of eastern Asia had more in common with the flora of eastern North America than western North America. In fact Asa Gray and Charles Darwin corresponded extensively in regards to this sort of plant distribution.
While much has been studied, researchers are still working on this problem to definitely decide how it happened. It certainly has something to do with widespread populations of species that then got separated, but the entire sequence of events doesn't seem to be completely mapped out yet. There are some nice explanations here and here.