'Seeing the elephant' is an English idiom with a slippery history. Its meaning has changed over time, but it is currently is used almost exclusively to mean that someone has seen active military duty and has experienced the horrors of war.

It appears to have gotten its start in the form of the British expression 'to see the lions'. In the late 1500s the Tower of London housed a collection of exotic animals, including a lion, an exotic and powerful emblem of the king. Many would come to London hoping to see the lion, and it became a cliche indicating that one had experienced the wonders of the big city -- however briefly.

Francesco was no other but a meere nouice, and that so newly,
that to vse the olde prouerbe, he had scarce seene the Lions.
-- Greenes Neuer Too Late by Robert Greene, 1590

Seeing the elephant, however, is specifically an Americanism. It originally indicated that one has seen something interesting, and perhaps has been underwhelmed by the experience. It could refer to adventures and travels, work experience, or simply an event that was notable. It often indicated a sense that the person was underwhelmed by the experience, or has become jaded. It seems to have been originally based on the excitement of travelling to a circus to see the strange animals, finding that they are indeed wondrous and strange, and realizing that you no longer have any need to see more.

"Now, Nick, don't hold him! Jist let the wild-cat come, and I'll tame him. Ned'll see me a fair fight won't you, Ned?"

"Oh, yes; I'll see you a fair fight, blast my old shoes if I don't."

"That's sufficient, as Tom Haynes said when he saw the elephant. Now let him come."

-- First American appearance in print, Georgia Scenes by Augustus B. Longstree, 1835.

Sometime around the American Civil War it started to become specifically a reference to war, a trend that continued through the Vietnam war. By the 60s and 70s it was used exclusively to refer to experiencing death and violence, with the understanding that this was a comparison to war -- and in fact, by far the most common usage was in reference to the violence of war. Since then usage has faded, but it is still around, primarily in military argot.

Igloowhite has proposed a different etymology in I glimpse the elephant, suggesting that it comes from the ancient Indian tale of The Blind Men and the Elephant. In this story a number of blind men encounter an elephant, but cannot agree on what it is, the one finding the trunk claiming that he had found a rope, the one finding the leg believing that he had encountered a tree trunk, etc. In this case, 'seeing the elephant' would be a reference to seeing a situation in its entirety, and understanding it fully.

I can find no evidence either confirming or contradicting this. However, An American Glossary by Richard Hopwood Thornton lists a number of examples of usage from 1840 to 1880, which can be found here. There are examples that seem to variously fit one or the other explanation, and I rather suspect that people have used the phrase with both etymologies in mind. However, I encourage you to make your own judgement.