It's slang, like "yeah" or "nah". They can be said to add emphasis. They are normally used as single word answers.

They're often used to give more emphasis. If you end the words with a clear 'p' sound, as opposed to an open 'o' or hissing 's', then it sounds more definite. If you're not sure you might use a the trailing-off sound of 'Yeahh, but .....' or 'Nah, I dunno...'. But if you're certain about your answer, you might give a smart, definitive 'Yep!' or 'Nope!'

"Why do people like to say...?" It's a slang form, but I think the "p" on the end gives a feeling of "finality". "Yep" = "Yes, and there's nothing more to add" or "yes, and that's that". Some people have also done the same thing to "well": "welp".

Thay are slang as Michael said. "Yep" can be said more quickly than "yes" because the final sound is the "p" which is short. The same with no versus nope, again we have the "p" sound. Yes is pronounced "yessss" Yep is pronounced "yep" The second is shorter and can be said slightly more quickly. Yep is more informal than yes.

It is fashionable.