Note: It's been a long time since I've seen The Hidden Fortress, but at the time I felt I wouldn't have recognized any similarity unless someone pointed it out. Star Wars certainly isn't a rip off of THF as others have claimed. Much of the similarities are part of any heroic fantasy epic.
Kurosawa himself was heavily influenced by John Ford. John Ford popularized many of the tropes we now associate with Westerns: sweeping vistas, riders on the ridge, lone riders, the reluctant scoundrel hero, etc... these also show up in Star Wars. Lucas would have also watched a lot of John Ford westerns making the whole question of influence circular.
That said, George Lucas was very influenced by Kurosawa and somewhat by THF. There's an interview of George Lucas talking about Kurosawa that I'd recommend you watch. He mentions at 0:53 that...
...my favorite of all time is really Seven Samurai. Then maybe Yojimo. And then Ikiru and maybe Hidden Fortress at that. So it's not at the top of my list but I was impressed and I liked it.
He borrows a lot of visual techniques from Kurosawa, here is a good video showing some of the more obvious ones. In particular, the opening of A New Hope is influenced by the opening of The Hidden Fortress.
He used a lot of camera techniques to get to the point. In the beginning of The Hidden Fortress, the first shot for example, the two peasants are panicked about something and you can't really see what it is. And the shot remains there as the horsemen sweep through it. And then you realize what's going on, and then they go away, and you're still sort of in the same frame.
Similarly, the opening of A New Hope is a static shot of a planet with the Rebel and Imperial ships entering the frame in combat. The long crawl of the Star Destroyer is very reminiscent of the stream of horsemen riding past the terrified peasants to demonstrate overwhelming power.
Music, framing and visuals tell the story. That is probably the most powerful influence from Kurosawa. If you watch a bad movie (not "so bad it's good") it will feature a lot of people talking about what's happening rather than showing you. It's cheap and it's lazy and it pads the movie out to 90 minutes. Kurosawa films are visual storytelling with the occasional important conversation to advance the plot. The first three Star Wars movies also work this way, there's very few scenes of just talking and they're short and to the point. George forgets that in the prequels and goes back to people walking and talking.
The only other key technique shared is the heavy use of wipes for scene transitions. This is now very rarely seen in US cinema and is a strong influence from Kurosawa.
As for plot and character elements...
Lucas recognizes that the strongest connection between the two movies is telling the story from the point of view of the two lowliest characters, the two peasants and the droids. 3:40 There's not a whole lot more connection between the characters than that. A pair of comic relief side kicks is a common trope, consider Rosencrantz and Guildenstern from Hamlet.
George says the princess trying to get through enemy lines is a coincidence and I agree. 4:07 Princess Leia derives her power and authority from herself; she gets them out of the prison bay, she flies the Falcon, she brings them back to the Rebel base. I don't think she ever refers to her royal position. The Princess in The Hidden Fortress acts more like royalty expecting others to follow her orders because she is the princess.