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I saw it in an email conversation at work (we work with americans). Someone sent this:

I asked if there was more information we could get through [...] and I was rebuffed as though I had lobsters crawling out of my ears

I googled it but there weren't any definitions, just instances where someone said it so I assume it's not defined anywhere on the Internet.

A Google Books search returns nine matches for "lobsters crawling out of [one's] ears"—all from books published during the period from 1986 to 2019. The impression I get from this surprisingly large number of matches is that the phrase may be idiomatic in some places, but it isn't an expression I've ever heard anyone use in the various parts of the United States and Canada where I've lived (Texas, Alberta, Maryland, Washington, D.C., New York, and California).

One of the oldest matches is from an unidentified article in Harvard Business School Bulletin (1986) [combined snippets], where the expression involves an abundance of actual lobsters at a dinner:

The weekend of 21 Sept. the Fishes and Kems spent a delightful two days with Ed White and Bernice in their beautiful new home on Chebeaque Island, ME. We became convinced that Ed and Bernice have accomplished near miracles in building and furnishing an island home when we listened to their stories of toting concrete transit mix trucks on barges and barging the many items needed for construction and furnishing. We very nearly had Maine lobsters crawling out of our ears when we partook of a delicious "White House" lobster dinner. Nothing like a feast of fresh caught Maine lobster!

But even older is an example that appears in Phil Grecian & ‎Jean Shepherd, A Christmas Story: A Play in Two Acts (2000) but actually goes back at least 17 years earlier:

RALPHIE. Schwartz told me he saw a pack of wild bears behind Pulaski's candy store last week. (He looks hopeful. MOTHER turns slowly to look at RALPHIE. THE OLD MAN lowers his paper.)

RALPH. They looked at me as if I had lobsters crawling out of my ears.

MOTHER (turning back to the stove). What would you like for Christmas, Ralphie?

As site participant Keep these mind points out in a comment beneath this answer, this play is based on the 1983 movie A Christmas Story, which itself contains the identical line about lobsters.

Similarly, no real lobsters are involved in the next-oldest Google Books instances (there are two), which appear in Kimm Walton, Guerrilla Tactics for Getting the Legal Job of Your Dreams (1995):

When I was in law school, about 10 years ago, if you had told me to do some soul searching about what kind of a job I wanted, I would have looked at you as though you had lobsters crawling out of your ears. I wanted what everybody wanted: a job with the biggest law firm that would take me. Did I do any research? No. Did I think at all about what would make me happy? Of course not. I figured that the jobs everybody else seemed to want would be fine with me, too.

...

To make this clear, imagine yourself walking into a car showroom. You've got at least some vague notion that you want to buy a car, otherwise you wouldn't be there. You stroll over to one of the cars and a salesman walks up to you. You say, "Tell me about this car," and the salesman says, "Well, it's pretty good for getting around, but the gas mileage stinks. You'd look at that salesman as though they had lobsters crawling out of their ears, wouldn't you? Of course you would! And that's because when you're buying something, you're also being sold something—and you'll only be sold on positives.

The meaning of "as though [one] had lobsters crawling out of [one's] ears" in both instances here is clearly along the lines of "as though [one] were plagued by some bizarre and unnatural affliction." The remaining six Google Books matches follow this same pattern.

For example, from ‎Jennifer Peterson, Beyond Perfect (2002) [combined snippets]:

"No, no. I believe you do. I just can't imagine a family with two masters of wit. I'll bet your family reunions are a riot."

"Actually, the rest of them spend most of their time gawking at us like we have lobsters crawling out of our ears. We don't blend in very well. But anyway, I am an only child, and I've lived here in Kansas for three years now." Gradually slowing her jog, she altered to a fast walk.

And from Debra Dirks & ‎Stephanie Parlove, Islam Our Choice: Portraits of Modern American Muslim Women (2003):

Finally, I was standing beside the Grand Imam Supreme with hundreds of people staring at me with such awestruck silence that I thought there must have been lobsters crawling out of my ears. Finally, I heard words slowly evolving. Were they coming out of my mouth?

The other four matches (from 2011, 2012, 2018, and 2019) are similar. On this record, it seems rather a stretch to suppose that the eight authors who use the expression idiomatically arrived at the same wording independently. In my view, they are invoking a shared idiom that is familiar (or at least known) to them—but not to most other people in the United States. Given that the earliest purely idiomatic usage is from 1983, the expression may have emerged fairly recently.

The phrase can be found in the book "The Forensic Comicologist: Insights from a Life in Comics" by Jamie Newbold. It can also be found at http://www.geocities.ws/p_cwick/Why_The_Fascination.htm

A Google search shows that there are other examples of usage of this phrase.

The phrase as it appears in both cases seem to suggest something extremely extraordinary and unlikely to happen, has happened.