近所のひとにカラオケに行こうと誘われました。
I can easily understand what this sentence wants to tell me, but I don't really understand why intentional form is used here. I usually would just use plain form いく.
The form with -ō isn't just for intentions (the intentional mood). It's also for invitations, i.e. an encouragement to do something together (the cohortative mood, like English "let's").
To understand the sentence, first we have to consider that the clause before と is a quotative argument. Xと誘われました means "I was invited to 'X'", where I put quotes around 'X' to emphasize how the particle と works; it takes complete sentences and treat them as quotations. That isn't done as often in English, but the structure is parallel to: "A neighbor invited me, like, 'let's go to the karaoke!'." – except in Japanese it doesn't sound casual or slangy; it's just a normal, unmarked way of expressing the idea.
You should parse it this way:
近所のひとに『カラオケに行こう』と誘われました。
(lit.) I was invited by my neighbor, (saying) "Let's go to Karaoke."
⇒ My neighbor invited me out to Karaoke. / suggested we go to Karaoke.
I usually would just use plain form いく.
You could rephrase the sentence as:
近所のひとにカラオケに誘われました。
I was invited to Karaoke by my neighbor.
⇒ My neighbor invited me out to Karaoke.
... but I can't think of a natural way to rephrase it using the plain form いく.
