Every model of a typed lambda calculus is a cartesian closed category.
Every cartesian closed category can be expressed as a typed lambda calculus (with the objects as types and arrows as terms).
Thus, typed lambda calculus and cartesian closed category are essentially the same concept.
It's a mistake (or at least imprecise) to say a monoid is a category with 1 element. But there is a canonical equivalence between monoids and 1-element (small) categories.
Recall that a monoid consists of a set $M$, together with a binary operation $\cdot$ and an element $e$, which satisfies the three equational laws $x \cdot e = x$, $e \cdot x = x$ and $(x \cdot y) \cdot z = x \cdot (y \cdot z)$.
Let's immediately note that for any (locally small) category $C$ and any object $A \in C$, there is a monoid structure on $Hom_C(A, A)$. The composition law is $\cdot = \circ$, and the identity is $e = 1_A$. So categories are filled with monoids. Call this monoid $M_{C, A}$.
On the other hand, for any monoid $M$, consider the category $C_M$ with one object $\star$, and with $Hom_{C_M}(\star, \star) = M$, with $1_\star = e$, and with $a \circ b = a \cdot b$ for all $a, b \in Hom_{C_M}(\star, \star) = M$. It's easy to verify the category laws on $C$ - they are just the three laws of the monoid.
And of course, you get that
$$M_{C_M, \star} = M$$
by definition. Furthermore, for any 1-element category $C$ where the 1-element is $\star$, you have
$$C = C_{M_{C, \star}}$$
again by definition.
This means that any theorem you can prove about all categories can be specialised to the case of 1-element categories, which can then become a theorem about monoids.
This is not to say that we "need" to represent monoids in this way. But it is a very helpful way to represent monoids.
Another way of taking a monoid $M$ and producing a category from it is considering the category $D_M$ where the objects are elements $m \in M$, and $Hom_{D_M}(n, m) = \{j \mid j \cdot n = m\}$, with the composition law being $\cdot$. It turns out that this representation can be obtained from the 1-element category representation $C_M$ using two general categorical constructions - the Yoneda functor and the "category of elements" - which you probably haven't learned about yet. So there isn't really a good reason to use this representation as far as I know.
Best Answer
I see that this question has an already accepted answer. Nonetheless I would like to give an alternative answer: sometimes seeing things from different perspective can help getting a better understanding of the subject.
There are many possible definitions/characterization of cartesian closed category (CCC in what follows), one of them is the following.
A CCC is a category $\mathbf C$ with:
If you reguard a category as some sort of generalized system of types and function between them then:
This parallelism makes arise a notion of interpretation of simply typed $\lambda$-calculus in cartesian closed categories: where an interpretation is nothing but a way to associate objects of a CCC to types and morphisms to $\lambda$-terms (functions).
Giving the details of this construction would be very long so I would rather avoid to give it here, by the way I think you can find different references on the subject: try googling categorical semantics of simply typed lambda-calculus.
I hope this helps.