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I basically agree with Grant's analysis, though I'm slightly more forgiving, I suppose. It's not *hysterically* amusing, but it's not where laughter goes to die, either. Mildly drole, perhaps.
Let's look at it with the stereotypical American tourist in mind:
The set-up: "Why do Americans speak so loudly?" = Americans talk _loudly_. The question purports to offer an explanation. (Perhaps we expect an answer like: Because stereotypical Americans abroad think everyone can speak some English and if you don't understand, talking at you more loudly will make it easier for you to understand. -- well, that's not funny by itself)
The punchline: Three things are going on: 1) The answer is not what we might expect. (humor). 2) wordplay with the word "loud" (humor) 3) reinforcement of the funny stereotype the question presupposes (humor).
"loud" clothes, as Grant explains, are overly patterned and colorful = garish, gaudy clothing. Stereotypical American tourists, particularly those on summer holiday, are thought to wear exceptionally gaudy clothing.
loud = high volume vs. loud = gaudy. Get it? There's the connector.
"Americans talk at too high a volume." and "Americans dress too garishly." By pretending to explain the first insult, the joke actually insults twice. 买一送二 Ha, ha, ha. :D
It's laudable that you want to distance yourself from this as potentially hurtful and ethnically insensitive, but as an American, I find this joke rather tame and inoffensive.
