Sure! Of course, it would be great if we could get a native speaker to just make sure that I haven't messed up on particles. I'm probably kind of rusty. But here's how it works.
The verb comes at the end. In this case, you use a humble way of saying "to speak," which is môsu, and it comes in the set phrase "môshi wake arimasen," which means, literally, "there is no reason for my speaking" or "there is no excuse for my speaking," and is used to mean "I am very sorry." So that's the "I am very sorry" part. (subject "I" is understood)
Then, the rest are attributes of the verb, so to speak (I'm not sure that's the correct technical term)
"urusai kaiwai ni makikomarete shimatte":<--nb, should have been SHIMATTE, not SHIMATE
"makikomu" means "to pull into, to entangle," and "makikomaru" is the passive "to be pulled into, to become entangled in" and "shimau" is a verb of finality and intensity. (When you add "shimau," it adds the sense of "and then this went and happened." It's the difference between "The car broke down [koshô shita]" and "The car went and broke down" [koshô shite shimatta]) So "makikomarete shimatte" intensifies and expresses a feeling of regret about the "become entangled in" "urusai kaiwa" means "unpleasant, loud conversation"
So, "urusai kaiwa ni makikomarete shimatte" means "become intangled in unpleasant conversation." The verbs are in a continuative form because they're joining up with your apology at the end. The apology has the main verb.
The rest of it is adjectives describing this type of American.
America-jin = American.
"sekai ni iyagarete iru yô na American-jin" means "the type of American who is despised/disliked by the world" (sekai = world) (iyagaru= to be despised/disliked) (yô na= type of, kind of, pattern of)
"Atama no karappo na" is an adjectival phrase, slightly slang-ish, meaning "empty-headed" "karappo" is slang-ish for empty. It has a connotation of a husk "atama"=head
so you run "Atama no karappo na" together with "sekai ni iyagarete iru yô na America-jin"
and you get "empty headed, despised-by-the-world type of American," which I'm translating as "empty-headed, universally despised type of American."
no subject
The verb comes at the end. In this case, you use a humble way of saying "to speak," which is môsu, and it comes in the set phrase "môshi wake arimasen," which means, literally, "there is no reason for my speaking" or "there is no excuse for my speaking," and is used to mean "I am very sorry." So that's the "I am very sorry" part. (subject "I" is understood)
Then, the rest are attributes of the verb, so to speak (I'm not sure that's the correct technical term)
"urusai kaiwai ni makikomarete shimatte":<--nb, should have been SHIMATTE, not SHIMATE
"makikomu" means "to pull into, to entangle," and "makikomaru" is the passive "to be pulled into, to become entangled in" and "shimau" is a verb of finality and intensity. (When you add "shimau," it adds the sense of "and then this went and happened." It's the difference between "The car broke down [koshô shita]" and "The car went and broke down" [koshô shite shimatta]) So "makikomarete shimatte" intensifies and expresses a feeling of regret about the "become entangled in" "urusai kaiwa" means "unpleasant, loud conversation"
So, "urusai kaiwa ni makikomarete shimatte" means "become intangled in unpleasant conversation." The verbs are in a continuative form because they're joining up with your apology at the end. The apology has the main verb.
The rest of it is adjectives describing this type of American.
America-jin = American.
"sekai ni iyagarete iru yô na American-jin" means "the type of American who is despised/disliked by the world" (sekai = world) (iyagaru= to be despised/disliked) (yô na= type of, kind of, pattern of)
"Atama no karappo na" is an adjectival phrase, slightly slang-ish, meaning "empty-headed"
"karappo" is slang-ish for empty. It has a connotation of a husk
"atama"=head
so you run "Atama no karappo na" together with "sekai ni iyagarete iru yô na America-jin"
and you get "empty headed, despised-by-the-world type of American," which I'm translating as "empty-headed, universally despised type of American."
And then "ano" at the very front means "That."
あの頭のからっぽな、世界にいやがれているようかアメリカ人のうるさい会話に巻き込まれてしまって、本当に申しわけありません。
I'll try to run it by someone Japanese on Facebook and get corrections.