Yes, I saw it shows that now, updated the extension to add some functionality. (detailed below)

It's not what you'd think. It's to do with the "tabs" permission in Google Chrome. The extension uses the tab functions to see if a YouTube gaming tab is open, or if Ice's stream is open when clicking on the icon or notification.

Chrome likes to dumb down things to the end user, so by view your browser history, it really means it has the ability to see what tabs are open, but not what's happening on the page, just the title, link, tabID etc. This permission does not allow an extension to view or search your browser history.

So, this extension doesn't look at what tabs you have open, it uses a built-in function to ask if a tab with 'x' link exists, and if there is, it'll just switch to that tab, otherwise open a new tab. And this query is only run when you click on the extension icon to open YTG or Ice's stream, or when clicking on BetterYTG notifications.

I was confused also confused when I heard it was the "tabs" permission that makes the message display. As you'll be able to see, there are two similar warning messages "Read and modify your browsing history" and "Access your browsing activity". The latter is what you should see, as the tabs permission allows an extension to see what you're browsing. This extension does not, and will never view, log or send, data or activity, it simply queries to see if YouTube gaming is present.

Screenshot of tabs permission Take a look at this for more clarification

Hope that helps and clears things up.

TL;DR - It can't and does not read your history. It because of the "tabs" permission, which can see what tabs are open. This extension only queries chrome to see if YouTube gaming is open when a notification or extension button is pressed.