Interpretation of regression coefficients

[ufc200sayef-oss]

The model shows a strong fit with an Adjusted R-Squared of 0.697. Interestingly, the coefficient for PROMO remains exactly -0.47751 both in the full model and in the partial regression of residuals. This numerically confirms the Frisch-Waugh-Lovell Theorem, proving that this coefficient represents the unique contribution of Promo to Price after removing the effect of Assor. Additionally, the very low p-values (< 0.001) confirm that both

[MarcoRianiUNIPR]

What you write is correct.
In multiple regression each coefficient $\beta_k$ represents the expected change in the response
variable $y$ associated with a one-unit increase in predictor $X_k$,
holding all other predictors constant.

More specifically:

The intercept $\beta_0$ is the expected value of $y$ when all predictors are equal to zero.
A slope coefficient $\beta_k$ measures the \emph{partial effect} of $X_k$ on $y$, controlling for the remaining predictors in the model.

For example, suppose the fitted model is
$\hat{y} = 3.2 + 1.5x_1 - 0.8x_2$.

Then:
Increasing $x_1$ by one unit increases the expected value of $y$ by $1.5$ units, holding $x_2$ fixed.
Increasing $x_2$ by one unit decreases the expected value of $y$ by $0.8$ units, holding $x_1$ fixed.

Geometrically, the regression coefficients define a hyperplane in the
predictor space. Each coefficient determines the slope of this plane
in the direction of its corresponding predictor.

If the predictors are correlated, each coefficient reflects a
partial relationship, that is, the linear association between
$X_k$ and $y$ after adjusting for the linear effects of the other predictors.

[ufc200sayef-oss]

appreciate your thorough and lucid explanation, Professor.
I like how the model accounts for these linear effects in order to determine the actual association.