I've a doubt about one kind of construction in linear algebra. First, I understand the definition of quotient space. Given the vector spaces $V$ and $W\subset V$ for all $x,y\in V$ we can say that $x\sim y$ if $x-y \in W$. With this we create the set of all equivalence classes denoted by $V/W$ and then we simply give it the structure of a vector space with convenient operations.
All of that is fine, however, I've seen already the use of this construction to "kill" elements. Like in the definition of tensor product, where we use this to grant linearity to the tensor product. I've heard that this construction is very common, and that it's one of the main uses of the quotient space.
How's that done ? Why does this construction kill the elements ? I've thought a little but I couldn't interpret it from the definition.
Thanks in advance for your help