For each asset $i$, consider the time-series regression to estimate $\alpha_i$ and $\beta_i$: $$ R^e_{it} = \alpha_i + \beta_i f_t + \epsilon_{it} $$

Take expectations to get rid of the error term and get:

$$ \operatorname{E}[R^e_{it}] = \alpha_i + \beta_i \operatorname{E}[f_t]$$

Meanwhile (*) says $\operatorname{E}[R^e_i] = \beta_i \operatorname{E}[f]$. This can only be true if $\alpha_i = 0$. If $\alpha_i \neq 0$ then $\operatorname{E}[R^e_i] \neq \beta_i \operatorname{E}[f]$.

For each asset $i$, consider the time-series regression to estimate $\alpha_i$ and $\beta_i$: $$ R^e_{it} = \alpha_i + \beta_i f_t + \epsilon_{it} $$

Take expectations to get rid of the error term and get:

$$ \operatorname{E}[R^e_{i}] = \alpha_i + \beta_i \operatorname{E}[f]$$

Meanwhile (*) says $\operatorname{E}[R^e_{i}] = \beta_i \operatorname{E}[f]$. This can only be true if $\alpha_i = 0$. If $\alpha_i \neq 0$ then $\operatorname{E}[R^e_i] \neq \beta_i \operatorname{E}[f]$. If you can reject $\forall_i \alpha_i = 0$ then you can reject the factor model that you're testing.

As a practical matter, the way these tests work out is that if you have portfolios or assets with any interesting and significant spread in expected returns then you can typically reject any asset pricing model. You can statistically reject the Fama-French three factor model. The more practical but fuzzy test is whether the model is telling you something useful about the data! Even though you can reject the Fama-French 3 Factor Model with the right test assets, that model captures something about the cross-section of expected returns that's empirically there while the CAPM is just entirely in tension with the data.