English represents statements indirectly by placing them in a subordinate clause beginning with "that": "I said that I love him." Here, the clause "that I love him" is the direct object of the verb of saying, that it, it is acting like a noun and it is called a noun clause. You can see how a clause can act like the direct object of a verb if you consider a sentence like "I said nothing." "Nothing" is the direct object of "I said." Compare: "I read the book" ("the book" is the direct object) and "I read that Aeneas founded Rome." ("that Aeneas founded Rome" is also the direct object of "read"; it is a noun clause and it represents Indirect Discourse.) Now literary Latin does not use a noun-clause to represent indirect discourse; it uses an Infinitive Phrase. This phrase is actually the direct object of the verb of saying. That is possible because Infinitives are nouns which represent a verbal notion without any limitations with regard to person or number. In Indirect Discourse in Latin the Infinitive must have a subject, and the subject must be in the accusative case. This Subject Accusative - Infinitive construction represents the Subject - Verb of a sentence and it acts like the direct object of the verb of saying. Dixi me eum amare = "I said that I love him. Consider the parallel: Dixi nihil.
Learn more about the origins of the indirect statement, here.