I assume that your main question is about how the residuals $e$ of the full and those of the restricted model $e_*$ can be used to derive the F statistic. The F statistics is used to test the null hypothesis that the (linear) restrictions applied to a set of regression coefficients hold true (for the population, the data generatig proces).

In your question you refer to the book of Green, which therefore I will do as well. The notation used in the answer conforms to Green's notation.

We have a linear regression model and we want to test some linear restriction on the regression coefficients, like $b_1=0$ or $b_1=b_2=0$ or $b_1=b_4=3$ or $b_1+b_2-b_5=2$, just to mention a few linear restrictions that could be tested.

We have a full model, no restrictions applied, the regression coefficients denoted by $b$. Also, we have a restricted model, the regression coefficients denoted by $b_*$. The residuals of the full model are denoted as $e$ and those of the restricted model as $e_*$. We'd like to use the difference of the sums of squares of $e$ and $e_*$ to obtain the F statistic. In vector notation, we are interested in using $e_*'e_* - e'e$ to obtain the value of the F statistic under $H_0$.

Green shows that the difference $b_*-b= -Cm$. , where $m = Rb-q$. $R$ is a vector or matrix specifying the restrictions (see Green) and $q$ is a vector containing values for the specified restrictions (also see Green). $C$ is a matrix:

$C = (X\,'X)^{-1}R\,'P$ where $P=[R\,(X'X)^{-1}R\,']^{-1}$. Also:

$C'= PR(X'X)^{-1}$, because both $P$ and $(X'X)^{-1}$ are symmetric.

The above formula for $C$ is arrived at by applying what Green describes in the paragraph "restriced least squares estimation". It's a pitty that the equation for $b_*-b= -Cm$ is not explained (or at least referred to) in the later paragraph where Green develops the F statistic in terms of the difference $e_*'e_* - e'e$ which we focus on in this question and answer.

Because $b_*= b-Cm$ it holds that $Y-Xb_*=Y-Xb-XCm$. Hence:

$e_*\,'e_* = (Y-Xb_*)\,'(Y-Xb_*) = (Y-Xb+XCm)\,'(Y-Xb+XCm)=$
$(e+XCm)\,'(e+XCm) = (e'+ m'C'X')(e+XCm)$

This can be elaborated further as:

$e'e+m'C'X'e \quad + \quad e'XCm + m'C'X'XCm = e'e + 0 + 0 + m'C'X'XCm$

And so we obtain:

$e_*'e_* - e'e = m'C'X'XCm$

Focusing now and $C'X'XC$ and substituting the expression for C given above, we get:

$C'X'XC = [PR(X'X)^{-1}]\,\, X'X \,\, [(X'X)^{-1}R\,'P]=PR(X'X)^{-1}R\,'P = PP^{-1}P=P$

Finally we get that:

$e_*'e_* - e'e = m'C'X'XCm = m'Pm=m'[R'(X'X)^{-1}R']^{-1}m$

or:

$e_*'e_* - e'e = (Rb-q)'\, [R'(X'X)^{-1}R']^{-1}\,(Rb-q)$

This final expression can be used to get the F statistic mentioned at the beginnig of this question.