This question already has an answer here:
Anyone who has studied quantum mechanics know the following relation: $ 2 \otimes 2 = 3 \oplus 1 $
But how did a man woke up and said "Hell yeah, I'll use tensor product of two spin $1/2$ to simulate the interaction of two particles with spin $1/2$" ? Why didn't he start with the direct sum ?
(And then, group theory made the magic leading to the relation above)
In fact, i'm wondering this because i don't fully understand why we use the tensor product to unite the two Hilbert's space.