I have difficulty in understanding few terms when i have to deal with direct product of groups. Let $G = A \times B$ be a group which is a direct product of two groups $A$ and $B$. It is easy to visualize the elements of $G$ as of the form $(a,b)$ where $a \in A$ and $b \in B$.
Question 1 : How to visualize $\text{Cent}_G(B)$?
Question 2 : How to visualize elements of $B$ in $A \times B$?
Question 3 : How to visualize $G/A$?
It is easiest to answer question 2 first. If we write $1_A$ for the identity element in the group $A$, then you can think of an element $b \in B$ as being the element $(1_A,b) \in G$.
Next, question 3 is related to question 2. It is a standard fact that $G/A \cong B$. The isomorphism is given by $\overline{(a,b)} \mapsto b$, where the "bar" denotes the equivalence class in the quotient. From the discussion in question 2 above, you can identify $a \in A$ with $(a,1_B) \in G$. Then you can see that $\overline{(a,b)} = \overline{(1_A,b)}$, since they differ by multiplication by the element $(a^{-1},1_B)$.
Lastly, let's look at question 1. The centralizer of $B$ is the set of all elements that commute with any element in $B$. Again by our previous discussion, we identify $B$ with $1_A \times B \subset G$. So a generic element in $B \subset G$ looks like $(1_A,b)$. If $(\alpha,\beta)$ is in the centralizer of $B$, then we have for any $b$:
$$ \begin {align*} (1_A,b)(\alpha,\beta) &= (\alpha,\beta)(1_A,b) \\ (\alpha,b\beta) &= (\alpha,\beta b) \end {align*} $$
You see that it doesn't really matter what $\alpha$ is. All you need is that $b \beta = \beta b$ for all $b$, or in other words, $\beta \in Z(B)$ (the center of $B$). So $\mathrm{Cent}_G(B) = A \times Z(B)$.