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Often said in poker when someone has a winning hand, as the others 'weep' over 
their loss. The expression can also refer to things that may cause distress to others.
* Well, would you look at that; read 'em and weep fellas, because I have a royal flush!
Note: The origins for most idioms are unclear. Often times, the origins you see listed are plausible theories to how an idiom came to be, but not necessarily so. The quotes you see that contain the phrase are the oldest that I could find, but it's very possible there are older recordings somewhere, so if you know of any, let me know!

Keep in mind, just because you see a saying in a newspaper from 1850 does not mean it originated in that year, or from that newspaper. In all likelihood, if a saying is already being used in a form of media like that, it's probably from an earlier time. The purpose of these old quotes is to show, with proof, how old some phrases go back in history.

Cards used to play various card games, like poker.
This phrase is believed to come from a popular card game known as poker. Poker is 
a game where players put together different combinations of cards that are worth a set 
amount of points. Certain arrangements of cards are worth more points than others, hence, 
during the game, players attempt to build the strongest combination of cards possible before 
the game is over. At the end, players reveal their cards to each other and whoever has the 
strongest set of cards wins.

Betting is sometimes involved when playing poker, which makes losing a bit more painful. Players who are feeling confident with the hand they have might, when it comes time to reveal one's cards, decide to place their cards on the and say something along the lines of 'read them and weep.' The player is essentially saying: "Look at the combination of cards I have, I've won."

Today the expression is used not just in card games, but for anything that may cause discomfort to another person. The earliest written form of the phrase I could find is only from the early 20th century. For example, the term is used under the 'Sports News' section of The San Antonio Evening News newspaper, printed in July 1919: 

    "With the exception of one or two close decisions, Umpires Lewis and Schaffer did faily good work, 
     probably because they had been made very good 'boys' by the irate fans . . . Few arguments were 
     indulged in with the arbiters. Read 'em and weep!"

Something interesting to note: Around the same time, in 1919, several newspapers used the idiom right before presenting a list of items for sale. The implication was that they probably expected customers to "weep" with happiness after seeing their cheap prices.
R-Letter Sayings  |  Know Your Phrase -  An Alphabetical List of Common Phrases, Sayings and Idioms With Their Meanings and Origins
A list of expressions starting with the letter - R.
The meaning of the phrase 'read them and weep.'
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Examples
Phrases and Idioms