Scrabble was invented in 1931, and the inventor was inspired by crossword puzzles (which were a massive fad during the Twenties... picture women wearing crossword-puzzle stockings). And the inventor of the crossword puzzle was, in turn, inspired by word square puzzles where the objective was to construct a grid of tightly interlocking words to form a square (or other shape) based on clued words. A typical word square looks like this:
C A R DIt is an interesting exercise to try to construct a word square,
A R E A
R E A R
D A R T
By specifying the letters in such a square, this can be turned into a relatively easy puzzle.
Word squares as puzzles date back centuries, but the word square formation has been in existence for at least two millennia. (See the palindromic SATOR square.) Finally, the Greeks are credited with the idea of ordering letters in lined-up rows and columns (c. 600 B.C.), though without any words running vertically.
These are some of the giants whose shoulders Bananagrams perches upon. Remember to thank them the next time you pray to the Bananagrams gods for good tiles.
Further reading: The Straight Dope on why crossword puzzles are symmetrical.