I'm absolutely not a specialist, but this is my contribution:

In your ANOVA model, you treated both 'recipe' and 'temperature' as fixed factors, which can be thought of in terms of differences.

In your linear mixed model, you treated 'temperature' as a random factor, which is defined by a distribution and whose values are assumed to be chosen from a population with a normal distribution with a certain variance. It turns out that the corresponding output is now an estimate of this variance (line labeled 'temperature' in the Random effects section). And you can notice that the output for the 'recipe' is indeed an estimate for mean-differences (lines labeled recipeB and recipeC in the Fixed effects section).

Very briefly: In a two factor ANOVA (or, more generally, in a model that can be analyzed with lm in R) variables are controlled for. That is, it asks "Holding other independent variables constant, what is the linear relationship of each independent variable with the dependent variable?" Such models have a number of assumptions, key here is that they assume that the errors (as estimated by the residuals) are independent. Often, this is reasonable; also, often, it is not. In the cake data set it is not, because each recipe is tested multiple times, and surely the errors from the model will be more similar within each recipe than across recipes.

Mixed models relax this assumption.