Although modern English is mostly an SVO (subject-verb-object) language, it still has some remnants of the original Proto-Germanic (SOV) word order. For other modern Germanic languages this is true to a greater extent, and they are generally described as V2 languages, V2 being an intermediate stage on the road from SOV to SVO. V2 means that a normal sentence starts with the sentence topic, followed by the finite verb in second position. (What happens after the finite verb is more complicated and differs between languages.) The sentence topic is either the subject or something you want to emphasize. In this case, the sentence topic is the direct speech: 'I will!', which grammatically is the object of the finite verb said.
English itself was a V2 language not too long ago, and the word orders in sentences such as "'Pop!' goes the weasel" or "Where is it?" are reminders of the fact.
The old V2 word order is gradually being replaced by SVO. Each generation uses V2 a little less and SVO a little more. The change doesn't happen uniformly. Contradicting the principle that pronouns simply replace noun phrases, today you are more likely to hear "'I will!' said the man" than "'I will!' said he". I believe the reason is that pronouns have a natural tendency to associate or even fuse with their verbs. (This is actually how personal verb endings - or in some languages personal verb prefixes - first arise. After the fusion, new pronouns are needed. E.g. spoken French has almost lost its personal verb endings but can almost be considered to have gained verb prefixes as in "j'entends" or "j'suis". And new pronouns are on the rise: "moi j'entends, moi j'suis".) Therefore, in some ways the combination of a pronoun with a verb behaves like a finite verb.
When V2 was still the norm, people found it slightly less wrong to say "Off it goes" than "Off the bomb goes" because smuggling the pronoun it into second position along with the finite verb goes feels like a less serious violation of V2.