Why does the F-ratio follow an F-distribution when the null hypothesis of equal grouped means is satisfied?

I'm really stuck on this.

I know that an F distribution is:

And the F-Ratio is:

MST= mean squares for treatment

MSE= mean squares for error

From here I'm totally stuck.

I think it all boils down to me not understanding why

Only when all of the group means equal each other.

It seems that

is always a chi squared distribution.