Iran advised the Houthi's not to take Sanaa. The Houthi's took it anyway when they found that it was virtually undefended. But for a long time they prefered to stay there and didn't make much effort to conquer the rest of the country.
The main focus of the Houthi's has always been Al Qaeda. One of their motives for their uprising was their perception that Hadi was supporting Al Qaeda and its hate campaign against the Zaydi's. The Houthi's only started their offensive towards the south after the murderous bombing of two Zaydi mosques in Sanaa.
The Zaydi's are "fiver" Shiites and in many ways closer to the Sunni than the Iranian branch of Shiism. They are estimated to constitute around 40% of the Yemeni population. Zaydi kings ruled over northern Yemen for over 1000 years. The Houthi movement consists mostly of Zaydi's but contains also some Sunni's.
Unfortunately for Yemen foreign interventions - always done with money rather than soldiers - are very popular in Saudi Arabia. And after the problematic outcomes in Syria and Iraq they are longing for a success story. Yemen looked like an easy target - and so the new king sought to raise his popularity by targeting it. It doesn't matter that the storyline of Iran-fueled Shiite aggression is fake: the Saudi public opinion loves it.
All signs are that the Houthi's are open to dialogue. But it is much less clear whether the Saudi's are.