It's a reference to the prophecy that King Arthur will return. The idea is that he was once king, and will be again.
As far as I know, T.H. White did in fact coin the English version of the phrase for his Arthurian book The Once and Future King, but you'll occasionally hear it adapted for other uses ("ladies and gentleman, the once and future champion!"), presumably as an allusion to the book. The original source is Sir Thomas Malory's Le Morte d'Arthur (the most enduringly popular rendition of the Arthurian myth), where the equivalent Latin phrase rex quondam rexque futurus is described as engraved on Arthur's tombstone.
When we deal with problems, we confront and attempt to solve them. Usually, we have to deal with things that are unpleasant. "Deal with it," is similar to "bite the bullet," "grin and bear it," or "face the music." In most situations, you can replace "Deal with it," with "Too bad you don't like like. Face it any way."
Here are some situations that should help give you some perspective on this:
A soldier complains to his superior about it being cold outside. "You don't like the cold, private? Too bad, you're a soldier--deal with it!"
A little boy complains to his mother about going to the doctor to get a shot. The mother says, "I don't care if you don't like needles. I don't want you getting sick, so deal with it!"
An employee says they don't want to open the store at 6am because they will be tired. The boss could reply: "You'll be tired? Oh, sorry to hear that, but your job requires it, so deal with it, or find a new place to complain."
If Donald Trump were to say that, he would mean "Too bad you don't like things but that's the way it is, so get over your feelings and confront the situation." The expression makes sense for Donald Trump because he's a deal-with-it kind of guy. There is a sense of harshness and matter-of-factness.
I wouldn't say it's an everyday phrase. It's certainly more common now (it would be interesting to trace it's roots.) It is widely known and accepted.
EDIT: I want to also add "Get over it," as another way to understand the expression. To "get over something," means to accept a situation and move on to the next issue. In my above examples, "deal with it" can be replaced with "get over it." without changing the meaning very much (if at all.)
Best Answer
This is based on a common type of phrase:
where the implication is that "If you bring me something that meets description A, I will be able to show you something that meets description B, because anything that meets description A also meets description B".
One example (from quick googling) is:
which means "All men with tattoos have interesting pasts."
So, from Trump's example we have that anyone who is a sore loser:
is also a loser:
Interpreting the phrase "show me a sore loser, and I'll show you a loser" is difficult, however. It could mean:
which implies "any failure is a failure, however you take it. i.e. any failure is bad".
Alternatively you could take Trump's phrase to mean:
Flipping this around, I interpret the phrase as meaning:
Side note:
My googling for similar phrases suggests that a much more common form of the phrase is "Show me a good loser and I'll show you a loser", which is pretty close to the first interpretation above of Trump's phrase. It implies that any failure is a failure, and there is nothing to be gained by learning to cope with failure.