Most gum is made from latex rubber, taken from the sapodilla tree of Central and South America.
"Soul" and "sole" are homophones, that is words that have the same pronunciation but different meanings or spellings.
1 a : the undersurface of a foot b : the part of an item of footwear on which the sole rests and upon which the wearer treads
2 : the usually flat or flattened bottom or lower part of something or the base on which something rests
Thus, "Rubber Soul" translates to "Gum Shoe"
Gumshoe is an early name for footwear that would come to be called "sneakers" because walking in these made practically no noise.
1 : one that sneaks
2 : a usually canvas sports shoe with a pliable rubber sole
For this reason a detective is referred to as a "gumshoe".
A detective, or gumshoe, examines clues.
Rubber Soul = Gum Shoe = detective ("stealth" is implied).