Sometimes the verb to mean is used as a synonym for "to have the effect, to cause ..."
Several more unexpected guests arrived, which meant we had to bring
out some more sandwiches.
That sentence could also be stated with "that":
Several more unexpected guests arrived, which meant that we had to bring out some more sandwiches.
That is, several more unexpected guests arrived, which had the effect of us having to bring out more sandwiches.
which refers to the stated fact that there were several more unexpected guests.
So, to apply that to your sentence
... widening partisanship, which had the effect of making readers more likely to believe stories that maligned people who belonged to the opposition party, or to believe stories that reinforced reader biases.
The idea behind partisanship is that people are "on different sides" of an issue. There is a gulf between them. "Widening" partisanship suggests that this gulf is getting wider, that is, the sides are moving away from each other.