The book is available at Internet Archive, Hathi Trust, and Google Books. There is also a free audiobook at LibriVox. You can find the stories listed and linked in Diigo (although there are no original stories in this book; they are all rhymed versions of stories published in prose elsewhere).
Brer Rabbit and the Tar-Baby
De Appile-Tree
De 'Gater and de Rabbit Gizzard
A Wishing Song
How Brer Tarry Pin Learned to Fly
"It's Good to be Old if You Know How to Do"
The Hard-Headed Woman
Two Tales in One—One Tale in Two
Why the Frog Has No Tail
Uncle Remus Captures a Dream
Why the Buzzard's Head is Bald
De Ol' Stand-Bys
Brer Rabbit's Gigglin'-Place
Mr. Rabbit Run Fur—Mr. Rabbit Run Fas'
Baylor's Mail
Revival Hymn
Camp-Meeting Song
Corn-Shucking Song
The Plough-Hands' Song
Christmas Play-Song
Plantation Play-Song
Transcriptions
I. A Plantation Chant
II. A Plantation Serenade
De Big Bethel Church
Time Goes by Turns
A Howdy Song