# Change in mean of y given change in mean of x?

I am a bit confused about interpreting simple linear regression coefficients. My understanding is that for a linear regression equation, increasing x by one unit corresponds to a change in the mean of y.

What I don’t understand is what is exactly meant by “mean of y.” It can’t be the arithmetic mean of the entire y dataset. Same for the x, I’ve heard the regression line passes through the mean of x and mean of y. How is this the case when different values of x will yield larger values of y (assuming positive slope)? How can it be a static number, or is it?

I am trying to get a better grasp of understanding what a one unit change in x corresponding to a k unit change in y really means because it has lost me at this point.