"multiplicative inverse in the modulo of the larger number" what does that mean?

The two numbers in his example are $4$ and $9$. The statement is that $4$ has a multiplicative inverse in the integers modulo $9$, or in other words, there is an integer $n$ such that $4 \cdot n \equiv 1 \mod 9$. The $7$ can be obtained by some trial and error (you only need to check the integers $1$ through $9$). He then gives an example of an integer that does not have a multiplicative inverse modulo $9$, namely $3$.

How familiar are you with modular arithmetic? What the author means is that if $\gcd(n,m)=1$ and $m<n$, then we can find a number $k\in\{1,2,...,n-1\}$ such that $mk\equiv 1(\mod n)$.

One way to find the multiplicative inverse is to use the Extended Euclidean Algorithm, but for something small like $4$ and $9$, it is pretty fast to just multiply $4$ by everything in the set $\{1,2,...,8\}$, and see what comes out to be congruent to $1$ modulo $9$. It is a fact from group theory that only one of these numbers should be the inverse.