The collocation of words you've drawn attention to comprise what is called a tongue twister (see here). Other examples include the following.
She sells seashells by the seashore.
Or
The glum groom grew glummer.
Or
Sneak thieves seized the skis.
Or
Peter Piper picked a peck of pickled peppers.
The primary purpose of a tongue twister is to entertain yourself and others. The standard method of doing so is by challenging someone to say the tongue twister, say, ten times in quick succession without stumbling on the words. The first person to do it successfully, wins.
As for your "rubber baby-buggy bumpers," for years a wheeled device for transporting babies, infants, and toddlers was called a baby buggy (or a pram, in the UK; pram being a shortened and altered version of the word perambulator]). The latest equivalent expression is a baby stroller. I'm assuming--perhaps wrongly--that the wheels on baby strollers today, as in days of yore, are still made of rubber, at the least the part that contacts the sidewalk or pavement.
If a baby buggy were to be equipped with a device similar to the front and/or rear bumpers on an automobile, and those bumpers were made of rubber, then I guess you'd have a baby buggy with rubber bumpers.
If you find a picture of a baby buggy with rubber bumpers, please let me know, as I'd be interested in seeing it!
Addendum:
Contributor Mari-Lou A referred me to Google Images, where pram + bumpers yields these pictures.