I purchased Minecraft and Xbox One for my kids to play together. After the initial installation, they were able to play. The second time they turned it on, it says they must sign in to Xbox Live. I don't have or want an Xbox Live account. Is this a requirement I was not made aware of when I purchased?
You can't really even use the Xbox One without an account signed in (you can really only do basic stuff, but once you try to do something it will ask you to sign in), so the first time they played, they were logged in. I'm guessing when you turned the system off it signed out.
You want to go to the circled tile and select a profile to sign in.
Account doesn't mean Xbox Live. You don't need Live to play, but you do need an account/profile
I can only give my experience from a Xbox 360, but I purchased Trials HD and Evolution from the Xbox Live store when I was a Gold Xbox Live member. My Gold status has expired, but I can still play as long as my Xbox is connected to the internet. There is no fee for this (just your ISP fee of course). You will need to create a Xbox Live account (probably already done if you bought the game). So I am thinking that you should just connect to the internet and will be good to go.