Gentzkow and Shapiro propose to measure the slant of a particular newspaper by searching speeches entered into the Congressional Record and counting the number of times particular phrases were used by representatives of each party, mechanically identifying phrases favored by one party over the other. For example, a Democrat is more likely to use the phrase "workers rights" whereas a Republican is more likely to use the phrase "human embryos". They then counted the number of times phrases of each type appeared in a particular newspaper to construct an index of the political slant of that newspaper.
The horizontal (x) axis is a subjective measure, where readers of the Mondo Times ranked newspapers according to their own beliefs about how liberal or conservative various newspapers are. Thus, newspapers are more conservative the further to the right and the higher they are on the graph. For the most part, the two measures largely agree with each other (for example, the Washington Times is both furthest to the right and third-highest from the top).
As for why newspapers take a particular slant one way or another, this study found that the most important variable is the political orientation of the people living within the paper's market. In other words, "papers to some degree are just giving their readers what the readers want so as to maximize the newspapers' profits."
For more on this study, see Econbrowser: What Drives Media Slant?.