In multiple linear regression why is it best to use an $F$-statistic when evaluating predictors?
The example given in your textbook proposes a multiple linear regression with 100 predictors, all of which have a "true" regression coefficient of 0. In other words, your independent variables have no statistical association with your dependent variable.

When you calculate the $p$-value of an individual coefficient, you're looking at the magnitude of the coefficient, the standard error of the coefficient, making some distributional assumptions about it, and asking the following question: "what is the probability of seeing a coefficient value this extreme if the true value is actually zero?"

If our distributional assumptions are correct, any given coefficient with a true value of 0 will report a &lt;0.05 $p$-value approximately 5% of the time. That's not so much of a problem if we only have one predictor, but by the law of large numbers, if we have lots of predictors, we'd expect 5% of them to report a $p$-value this low. This makes the $p$-values in high-dimensional regressions hard to interpret.

The $F$-test is different. Instead of evaluating every single coefficient for statistical significance, it applies a single test to the entire regression. So instead of having 100 chances to throw up an erroneous $p$-value, it only has one chance. This makes the $F$-test useful for evaluating whether or not there is a regression effect for high-dimensional regressions.