Why use an ANOVA/LRT to test for significance of a factor? I've been asked to use a likelihood ratio test to test for the significance of a main effect in a linear model. I have done so as follows:
anova(model1, model2)

Model 2 contains one fewer independent variable than Model 1.
The P-value obtained from the ANOVA is exactly the same as the P-value for the dropped term in the summary for the full model (Model 1). Why then is it useful or important to use an ANOVA to determine the significance of an effect? The only reason I can think of is when the effect is a categorical variable of more than two levels.

ANOVA, strictly speaking, does not assess the significance of a single effect, or a single predictor. Rather, it compares nested models, as you see in your call to it. Specifically, the larger model will of course always explain more variability in the data than the smaller one, and ANOVA tests the null hypothesis that this additional variance explained is consistent with no additional explanatory power of the larger model over the smaller one.
In the special case where the larger model has exactly one more degree of freedom than the smaller model (e.g., if the two models differ by exactly one numerical predictor, or by a binary categorical one), you are right that we could have gotten the exact same information from the t statistic of the estimated parameter and its associated p-value. (If you look at summary(model1), you will notice that the square of the t statistic is equal to the F statistic in the ANOVA table.)
However, the two models might differ by more than one degree of freedom, e.g.:

*

*If they differ by a categorical predictor with more than two levels, as you note.

*If they differ by a numerical predictor that is modeled with more than one degree of freedom, e.g. because it is spline- or polynomially transformed.

*If they differ by an interaction of predictors, some of which fulfill one of the two bullet points above.

*Finally, we may "intrinsically" want to compare models that differ by multiple predictors. For instance, we might be interested in the impact of weather on our DV, so the smaller model might contain many predictors, but not the weather, while the larger model contains multiple weather-related predictors, like the daily maximum and minimum temperature, hours of sunshine, rainfall quantity etc. If we are interested in "weather in general", it makes little sense to look at the t statistics of all these separate variables in isolation.

