It's a sarcastic response.
Taken literally the sentence would mean the speaker isn't already familiar with the situation and wants more information. However it is rarely (these days never) meant literally, and is used sarcastically to mean the opposite. I.e. That the speaker is already familiar with the situation. Its meaning is similar to the expression, "You don't say?", where the speaker is acting like they don't already know, when in fact they do.
The difference between these two expressions however is that, "Tell me about it", has taken on a sympathetic tone over the years, indicating shared misery, whereas, "You don't say?", is an insult where you don't actually care what the person has to say.
There's little in the words themselves to indicate that difference. That difference is simply the connotations those expressions have taken on over time. Tone of voice is pretty critical here, because the difference between a sympathetic statement and an insulting one is simply a different inflection while saying the exact same words.
The Chamber's dictionary was taking you in the right direction.
The phrase, "speaks to one’s heart," has a long history in the Western Christian experience and high currency in today's religious discourse. The idea of "speaking to the heart" appears in the Bible repeatedly. See for example God speaking in Ezekiel 3:10
"Son of man, take into your heart all My words which I will speak to
you and listen closely."
Presently, there is an extremely popular song/hymn that moves across Protestant churches and takes the phrase as its title. Just a sample of the lyrics:
"Speak to my heart, Holy Spirit Message of love, love to encourage me"
Now compare that with a 18th century Bible commentary/concordance when defining the phrase:
"speak to one's heart, to comfort him, to say pleasing and affecting
things to him."
That same definition runs verbatim through various other concordances and in the mid 19th century Noah Webster picked it up and (undoubtedly encouraged by Transcendentalism) put it in the current words of the hymn:
"To speak to one's heart, in Scripture, to speak kindly to ; to
comfort; to encourage."
By now you my have noticed that the phrase "Speaks to one's heart" is not synonym with "to touch one’s heart," though they are closely related.
An exegesis of the biblical texts on this subject reveals the principal issue is the passing of information, but not any kind of information nor in any kind of language.
The verb "speaks" plays a crucial role. While "touching" is undefined, subjective and open to various interpretation, "speaking" is about connecting to the recipient with words that make sense either intuitively or by logic. The point is that whatever "speaks to one's heart" knows the language to convey the meaning and also knows (and this is as important) the information that the recipient needs to "hear."
Two important things:
1- The language (which includes the rapport and manners: treats you with respect and speaks your tongue).
2- The message (exactly what you need to hear [film "The Matrix]).
The result of "speaking to one's heart," however, is the same as "touching one's heart." The person is reestablished, energized, and reoriented (with all of its geographical implications).
In short, when something "speaks to your heart," it fills the frontal lobe of your brain with essential information and then reaches to your amygdala with tenderness.
Best Answer
The toast goes back more than half a century before the scriptwriters of Casablanca used it in 1942.
From Anonymous, A Holiday Skip to the Far West (1884), we have this scene set in the Southern Hotel in St. Louis, Missouri:
From "Things Pleasant and Otherwise," in Ballou's Monthly Mahgazine (May 1884):
From "Editor's Table" in The Yale Literary Magazine (April 1895):
From Edith Chase & W.E.P. French, Waes Hael: A Collection of Toasts Crisp and Well Buttered (1903):
Early extensions of the toast include this one from The Harrow (1889) [result not shown in snippet window]:
And this one from Wilbur Nesbit, The Loving Cup: Original Toasts by Original Folks (1909):
I imagine that in a heavy-drinking, toast-oriented social environment, few dedications would come to the befogged mind more readily than "Here's looking at you," since the person addressing the toast is in all likelihood looking at the person thus addressed.
An early filmed instance of the expression occurs in the movie Three on a Match (1932), as Michael Loftus (Lyle Talbot's character) attempts to woo/seduce Vivian Kirkwood (Ann Dvorak):
Evidently, the expression was widely viewed as being quintessentially American, which would make sense in Casablanca as another marker of Rick's Americanness. At any rate, in one of J.P. Marquand's Mr. Moto stories (by 1938), we have this bit of dialogue [combined snippets]:
UPDATE: 'Here's Xing' as an American form
Because so many readers have echoed the comment/question (from andy256 below) "Yes, but ... what does it mean? Or if it is quintessentially American, what does that mean about America?" I decided to look into the idiomatic use of "Here's Xing" in U.S. usage.
Since Americans are (as Fowler says of people who grew up with the distinction between will and shall) "to the manner born," the form "Here's looking at you" doesn't strike us as particularly odd. Indeed, the similarly formed expression "Here's hoping" is used not just as the start of a toast, but as an equivalent (in a multitude of settings) to "I hope so."
One of the earliest examples of "here's hoping" in a Google Books search appears in "Metaphors of the People," in The Galaxy (May 1870), which quotes Mark Twain's description of toasts given in various locales in the U.S. West:
Of all these expressions, only the "Here's another nail" and the "Here's eighteen hundred barrels" follow (arguably) grammatical form, but it appears that the "Here's Xing" wording arose fairly early and has shown impressive cultural persistence. In fact, it appears in James Fenimore Cooper's The Pioneers (1823):
Given decades of "Here's hoping..." in the context of toasts and in the ordinary affairs of life outside barrooms, the emergence of "Here's looking at you" can hardly have struck listeners as a strange innovation. That toast, in turn, may have prompted the toast, "Here's mud in your eye," which a Google Books search finds as far back as 1912 in the form, "Here's mud in your eye and a quick trip back to Texas!" There are numerous competing interpretations of that expression, however.