I'd say what is confusing about the definition above is that it doesn't make clear that the binary operation is part of the monoid - it only asserts the existence of the operation on the set. For example, the above definition would make $\mathbb N$ a monoid because there exists an associative binary operation blah blah blah. But there are many such associative binary operations on $\mathbb N$.
It should really say, "A monoid is a pair $(M,\star)$ where $M$ is a set and $\star$ is an associative binary operation $\star:M\times M\rightarrow M$, such that there exists an $i\in M$ statisfying $i\star m = m\star i = m$ for all $m\in M$.
In particular, when the definition above says: $Ia=aI=a$, that is shorthand for the operation $I\star a = a\star I = a$.
Oh, and the only reason we tend to write monoids in a "multiplicative form," rather than more like addition, is that addition, in almost all instances, is commutative: $a+b=b+a$. But multiplication in many instances is not - for instance, matrix multiplication is not commutative. So we usually think of the monoid operation as being "like" multiplication.