Say you have a dataset with 2 features, $X$, and $Y$.
Centering the features of this dataset would mean we make it so that the mean value of the new $X$ feature becomes $0$, and the mean value of the new $Y$ feature becomes $0$.
In other words, if we get a dataset where the $X$ and $Y$ feature-values are centered, that means that they're given in terms of the distance from the most average feature-values for that dataset. When $X$ is $0$ and $Y$ is $0$, that corresponds to the "most average" datapoint.
Now, let's say we want to predict $Y$ based on $X$ by doing linear regression on this data-set. In other words, we are turning $Y$ into the label we want to predict, and $X$ into the single feature we're predicting it with.
Now, what happens? If we don't center either of the random-variables, then our linear regression model will simply fit a line to the $Y_s$ based on the $X_s$. $$y=mx+b$$.
That's okay. However, in this case $+b$, the intercept, has absolutely no meaning. It's the label which our model predicts a datapoint would take when $X=0$. However, $X$ might correspond to a feature where equaling zero is absolutely senseless! For example, what if $X$ is the height of a newborn baby?
By centering $X$, the bias now has meaning. Since $X$ is centered, it now takes on a value of $0$ for its average value. That makes the bias, $+b$, equal to the $Y$ feature-value for the baby with "the most average height!!!" Pretty cool!
Now, what if we center the $Y$ labels as well? What happens to the bias - what does it represent?
First of all, notice what happens to the linear if we center the $Y$ labels. Centering the $Y$ labels means that we subtract the mean of $Y$ from each $y$ value, so that the value that was previously the mean.
Let $X'$ be the centered feature, $Y$ be the uncentered label, and $Y'$ be the centered label.
If all we're doing is setting $Y'=Y-\mu_y$, then our linear equation should go from...
...that is, all that should happen to our linear equation is that it should get shifted down by $\mu_y$.
Say that $Y$ corresponds to the weight of a newborn baby.
Before centering $Y$ and turning it into $Y'$, when $X'=0$ the linear equation told us what we should predict the weight of a newborn baby of average height to be. That was $b$.
Now, when $X'=0$, we instead get $b-\mu_y$. That is, we get the expected difference between the weight of a newborn baby of average height and the average weight of these babies!
Now, for each value of $X'$, what we're getting is the expected difference between the weight of a baby whose height is $X'$ away from the average height and the weight of the baby with the average weight!