An excellent question and in a rather confusing lesson I attempted with my students I think we discussed a very reasonable answer.
Imagine you have a bunch of data points that you would represent as ordered pairs: (x1,y1) (x2,y2) so on and so forth. We want to standardize these points and therefore take the z-score of each coordinate and then replot these z-scores. (Zx1,Zy1) (Zx2,Zy2) so on and so forth.
Our new plot is now "centered" on the coordinate plane. More importantly (xbar,ybar) is now (0,0), this comes naturally from the definition of a zscore. Z=(x-xbar)/s If x=xbar naturally our zscore becomes 0.
Now, I'm an AP Statistics n00b and I would personally stop here and say the least-squared line of z-scores always passes through the origin but that may seem a little like "hand-waving" to your students so you can continue.
The equation for a linear model is given by yhat=bnot+b1x
bnot=(ybar)-(b1)(xbar) where b1 is the slope. TADA!!! with our standardized scatter-plot both xbar and ybar are 0 and therefore the y-intercept of our standardized regression line is 0. Therefore, our regression line must pass through (0,0) AKA (xbar,ybar).