In the text it states that:
Definition 7.2. Let $\mathfrak{C}$ be a category and $\{A_i | i \in I\}$ a family of objects of $\mathfrak{C}$. A product for the family $\{A_i | i \in I\}$ is an object P of $\mathfrak{C}$ together with a family of morphisms $\{\pi_i : P \rightarrow A_i | i \in I \} $ such that for any object $B$ and family of morphisms $\{\phi_i : B \rightarrow A_i | i \in I \} $, there is a unique morphism $\phi: B \rightarrow P $ such that $\pi_i \circ \phi = \phi_i$ for all $i \in I$.
I could not get the part that is defining a "product" as an object of the category. The definition of category is very clear at the beginning of this chapter and it states that $\mathfrak{C}$ contains only the objects however here the "products" also included in it. Is this part of the definition true?
I suspect the confusion here is caused by the use of "together with". Beyond what has already been said in the comments, the following may help.
To keep things simple, suppose $I=\{1,2\}$. Then a product of objects $A_1$ and $A_2$ in $\mathfrak{C}$ is a diagram in $\mathfrak{C}$ of the form
$$\require{AMScd}\begin{CD} A_1 @<{\pi_1}<< P @>{\pi_2}>> A_2\tag{1}\\ \end{CD}$$ such that for any diagram in $\mathfrak{C}$ of the form $$\begin{CD} A_1 @<{\phi_1}<< B @>{\phi_2}>> A_2\\ \end{CD}$$ there's an arrow $\phi:B\to P$ unique making this diagram commute in $\mathfrak{C}$: $$\begin{CD} A_1 @<{\phi_1}<< B @>{\phi_2}>> A_2\\ @| @VV \phi V @|\tag{2}\\ A_1 @<{\pi_1}<< P @>{\pi_2}>> A_2 \end{CD}$$ The product is not just the object $P$, it's the whole diagram (1), but we often informally say "product $P$" and may even think about it as the object $P$ in $\mathfrak{C}$ when the arrows $\pi_1$ and $\pi_2$ are understood to be "together with" it.
The product (1) is actually an object in a different category, namely the category $\mathfrak{P}$ of "pairs of arrows in $\mathfrak{C}$ from an object into $A_1$ and $A_2$". This category consists of the following data:
A product (1) in $\mathfrak{C}$ is precisely a terminal object in $\mathfrak{P}$.
What "is" a diagram as an object in $\mathfrak{P}$ "really"? We could think of a diagram like (1) as a triple $(P,\pi_1,\pi_2)$. Note we need to keep track of the domain and codomain diagrams for the arrows in $\mathfrak{P}$, so an arrow like (2) could be thought of as a triple $(\phi,(B,\phi_1,\phi_2),(P,\pi_1,\pi_2))$.
I should probably clarify that none of this tells you that a product actually exists for objects in $\mathfrak{C}$, because a terminal object may not exist in $\mathfrak{P}$. For example, any preorder can be viewed as a category in which the objects are the elements of the preorder and there's a (unique) arrow $a\to b$ if and only if $a\le b$. In such a category a product is precisely a greatest lower bound, which need not exist. Other examples can be found in other answers on this site.