Why a monoidal category with only one object is a monoid in the category of monoids?

The only object in a monoidal category with one object is the unit $I$. Thus the only morphisms are those from $I$ to itself. These include the identity morphism and are closed under the associative composition, so form a monoid by definition. Now the monoidal structure necessitates that $I\otimes I$ is the same as $I$. Consequently, given two morphisms $f,g\colon I\to I$, you can interpret $f\otimes g\colon I\otimes I\to I\otimes I$ as a morphism $I\to I$. Thus, $\otimes$ gives a second way of combining morphisms $I\to I$. Moreover, the properties of $\otimes$ require that identity morphisms are a unit for this way of combining as well, and that it distributes over the first way: $ff'\otimes gg'=(f\otimes g)(f'\otimes g')$. This is the "monoid in the category of monoids" that is meant.