They like people who attack Obama and other Democrats because division in the Democratic party helps them win elections -- it was and still is all about taking down the Democrats, which is why they often promote "liberal" critics of said party, especially when they share similar prejudices.
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"The latest Democratic candidate to enter the 2020 race has an unexpected base of support: The far right and conservative media.
Rep. Tulsi Gabbard (D-HI), who announced her candidacy last Friday, has cultivated a fandom among the right by bashing fellow Democrats and espousing views that break with the party line...
Gabbard first became an in-demand Fox News guest in 2015 after she criticized Barack Obama’s unwillingness to use the label “radical Islamic terrorism.” Her media tour explaining that position earned her positively-tilted coverage in right-wing outlets like Breitbart and The Daily Caller—a trend that continued when she later expressed skepticism of Obama’s Iran nuclear deal.
One person with direct knowledge told The Daily Beast that in the wake of her Obama criticism of Obama, Gabbard became an increasingly requested guest for Fox News hosts and producers to appear on-air. They weren’t the only ones in television news who took notice: senior executives at Sinclair Broadcasting made appeals for Gabbard to appear on their networks after she rebuked Obama.
And her emergence as a left-wing Obama critic further put Gabbard on the map in conservative media."
https://www.thedailybeast.com/why-co...-for-president