Most of the Story Library screen is taken up by a list of the stories in your library. There will be a card for every story in your library, with a small visual preview of its structure1, how many passages it has, and when it was last changed.
To select a story, click or tap on it. You can only select one story at a time. Once selected, you can take action on the story using the top toolbar.
By default, Twine will sort your library by story name, so that a story named "Aardvark Revenge" will appear before "Zebra Brigade." You can instead have it sort your stories so that the most recently-edited ones appear first by choosing Sort By under the View top toolbar tab, then Last Updated. Choosing Name under the same button will change back to sorting by name.
You can tag stories to organize related ones together. Once you've tagged a story, you can filter the Story Library screen to show only stories with certain tags.
To do this, choose Show Tags under the View top toolbar tab. Choosing a particular tag will cause only stories with that tag to be shown. You can select multiple tags from the Show Tags button. In that case, Twine will show any story that has a checked tag name in the list.
To go back to showing all stories in your library, choose Show All Stories from the Show Tags button.
If the Show Tags button is disabled, that's because you haven't tagged any stories yet.
The color of your story preview is more-or-less randomly governed by your story's name. There isn't a significance to the colors--they're just there to help you find a particular story faster, and to look pretty.