Without getting caught up in a detailed discussion of the particular tendencies of left-wing, right-wing, and centrist politicians in the United States, I think you can make sense of the quote in this way: In the United States, "left" means (to some degree) liberal, while "right" means (to some degree) conservative. Democratic candidates, in general, run to the left of Republican candidates in their districts; that is, their positions on various political issues tend to be more liberal than their Republican opponents'. Conversely, Republican candidates tend to run to the right of Democratic candidates in their districts; that is, their positions on various political issues tend to be more conservative than their Democratic opponents'.
The problem that the Washington Post is raising is the prospect that a Democrat who supports an attack on Syria may be inviting a challenge in the Democratic Party primary for his or her district from a Democrat who is more strongly left-wing (the unspoken assumption here is that a more left-wing view is associated with a pacifist attitude toward Syria). That challenger would be said to be "running to the left" of the incumbent Democrat.
In recent years a similar difficulty has plagued incumbent Republicans who have favored compromising with Democrats on (for example) raising the nation's debt ceiling. In that case, Republican incumbents have sometimes drawn opposition in Republican Party primary elections in their districts from other Republican candidates who "run to the right" of them (where being "right-wing" is equated with being hostile toward government indebtedness).