Official answer:
Conclusion: "My brother stole a cake I had made." This is the speaker's main point. Why does the speaker believe that his brother is the culprit? The premise will make this clearer.
Premise: "He had icing on his fingers, and he said he wasn't hungry when I asked him whether or not he wanted dinner." The two phrases bolster the speaker's conclusion. If the brother did in fact eat the cake, it could reasonably follow that he would have icing on his fingers and that his hunger would have subsided.
Assumptions: There are numerous assumptions here—some more pertinent than others. For example, we assume that the brother likes cake, and we assume that there was not more than one cake for him to eat. Two important assumptions pertain directly to the premise: 1. Eating a cake puts icing on your fingers / the brother could only have gotten icing on his fingers from having eaten a cake; 2. Eating a cake makes you full / The brother got full from eating the cake (and not from eating something else).
Random details / Background information: "It was Sunday afternoon" and "I was angry." Neither of these statements constitutes premise. They have nothing to do with the speaker's assertion that his brother ate his cake. They could become premise if there was a statement like "My brother always eats cake on Sundays"; however, since no such claim is made, these facts are irrelevant.
Choice A rephrases the premise about the brother not being hungry. It does not effectively strengthen or undermine the conclusion.
Choice B strengthens the icing premise by making it less likely that the icing got onto the brother's fingers by accident.
Choice C is an irrelevant topic. We are not concerned with the deliciousness of the cake.
Choice D effectively undermines the icing premise by discrediting the assumption that links the icing on the hands with the eating of the cake.
Choice E shifts our scope from who ate the cake to what type of cake that person ate. This choice is also too narrow to be useful, since we do not know what kind of cake the speaker made.