Do I accept or reject the null hypothesis? M1 : Y ∼ β0 + β1x1 + β2x2
M2 : Y ∼ β0 + β1x1
anova(M1,M2) shows a p-value of 0.0001, so we
prefer M1 at significance level 0.05.
Would that be correct? I thought that if .0001<.05, I should reject the hypothesis M1?
 A: Your confusion stems from the lack of clarity about the null hypothesis that is being tested. P-values should always be interpreted taking into account the null hypothesis.
When we compare two models using anova(M1, M2), we are performing a likelihood ratio test with the null hypothesis: is the extra parameter in M1 ($\beta_2$), when compared to compared to M2, equal to zero?
If you reject the null hypothesis when the p-value is 0.0001 < 0.05, you can state that there is enough evidence to say that the extra parameter $\beta_2$ in M1 is non-zero. In this way, you will prefer M1 instead of M2. Otherwise, you would miss the explanation of $Y$ given by $X_2$.
One additional detail is that we never accept a hypothesis. The absence of evidence is not evidence of absence. You can read more here. For example, if you had observed a p-value > 0.05, then you would only be able to state that there is not enough evidence that the parameter is not zero (not rejecting the null hypothesis), but you could not say that the parameter is zero (because that is accepting the null hypothesis).
