"I eat breakfast."
を is used to indicate the direct object of a sentence; i.e., the thing the verbs affects, in this case the breakfast (あさごはん) that's being eaten. は defines the topic of a sentence, as in the general thing you're talking about and can often be translated as "regarding" or "as for." If you were to use は instead of を in this sentence, it would mean "the breakfast eats," which is obviously not right.
There is a more formal name for each of the meals as well, however duolingo focuses mainly on the less formal (asagohan, hirugohan and bangohan) . Which all share 'gohan', meaning 'meal', or 'rice'. The more formal names are: choushoku, chuushoku, yuushoku, which all share 'shoku', from 'shokuji', also meaning 'meal'... Confusing huh!