Given a monad $T: \mathbf{C} \to \mathbf{C}$ with natural transformations $\eta: id \Rightarrow T$ and $\mu: T^2 \Rightarrow T$, I want to construct a new category $\mathbf{C}^T$ where objects are the same as $\mathbf{C}$ and the hom-sets are $Hom_{\mathbf{C}^T}(A,B)=Hom_{\mathbf{C}}(A,T(B))$ for objects $A, B$.
However, I'm not sure what composition in this category should look like. If I have $Hom_{\mathbf{C}^T}(A,B) \ni f: A \to_T B$ where subscript $T$ denotes arrows in $\mathbf{C}^T$, then this is given by a morphism $f: A \to T(B)$ in $\mathbf{C}$. If I have $g:B \to_T C$ as a morphism $g:B\to T(C)$, I want to construct a composition $g \circ f :A \to_T C$ as a morphism $A \to T(C)$. But there seems to be no way to do this since there's no way to specify a morphism $T(B) \to B$ that we can use to compose with in $\mathbf{C}$.
Applying $T$ to $g$ you get $T(g):T(B)\to T(T(C))$ which you can compose with $f$ to get $T(g)\circ f:A\to T(T(C))$. Then you can compose with $\mu_C:T(T(C))\to T(C)$ to get a morphism to just $T(C)$.
(Incidentally, it is worthwhile to think about questions like this with some simple examples in mind. For instance, if $T$ is the free monoid monad on sets, you're asking for a way to compose "maps" $A\to B$ which instead of giving an element of $B$ for each element of $A$ give a string of elements of $B$. Given two such "maps" $f:A\to B$ and $g:B\to C$, how would you define $g(f(a))$? Well, $f(a)$ is a string of elements of $B$, so you can apply $g$ to each one of them to get a big string of elements of $C$. If you unravel what this means in terms of the monad structure of $T$, you get the construction above.)