Why is free monoid called Free?

Why in the world do they use the word "free" in "free monoid"? It driving me crazy to see where the "freedom" comes from.

Here is the Awodey's explanation of it, in terms of "baby lagebra" (sic.) but it is even more confusing:

A monoid M is freely generated by a subset A of M, if the following conditions hold

1. Every element $m\in M$ can be written as a product of elements in A:
   $m = a_1 \cdot_{M} ... \cdot_{M} a_n, a_i\in A$
2. No "nontrivial" relations hold in $M$, that is, if $a_1...a_j = a\prime_1 ... a\prime_k$, then this is required by the axioms for monoids.

to me this doesn't explain the word "free"...

Math level: novice

You are an element of a monoid, say $x$. You want to strike out on your own, you want to act on another element $y$ and be a unique individual, not conforming to the laws of monoid society. But alas, the law says that the relation $xy = e$ holds; so when you act on $y$, you can't express yourself uniquely...you are only the identity element $e$ =(. You yearn for freedom,but you have been chained down by the tyranny of the monoid relation $xy = e$.

The above is an example of a monoid that is not free. Intuitively, a monoid is called free if, as you mentioned in your definition, there are no relations, i.e. equations, that relate the elements together, other than the conditions (axioms) that all monoids must obey. When there are relations, that means that the elements of the monoid must also obey additional constraints, and you can interpret this as being like a loss of freedom.