Yes the Houthi movement are in control of large swathes of Yemen. Basically the US started drone bombing the hell out of Yemen back in the 2000s and what always happens happened - it drove way more people into a group that was opposed to the West.

As for the topics of the article, here's a mainstream news article that covers some of the same ground, and here's a rightwing news source. Join the dots, the main topic of OPs article is credible even if you have reservations about the tone.

(The famine in the last article has been going on for the better part of a decade without western powers caring so Sky's very hypocritical.)