The history of emoji ties back to the Japanese cellular market in the pre smartphone era. Texting using the Japanese language on a T9 keypad wasn't a great experience. Emoji emerged from one of their phone companies as a way of sending quicker text messages. The idea was to expand the range of ASCII emoticons/smilies widely that were showing up online. Each provider ended up creating their own emoji and they weren't compatible between networks initially. Eventually standards came into play, and I think around 2010 is when Unicode came in to the emoji picture.

The iPhone does have some ties to this becoming used worldwide. Initially Apple added emoji support to the iPhone when they were focused on the market in Japan. It was essentially an informal requirement there due to how popular emoji became on older phones. People outside Japan were finding ways to enable emoji and noticed it worked iPhone to iPhone at the time. Android adopted emoji as well, standards came into play, and it's now used around the world.

Here's the Unicode 8.0 document on "Miscellaneous Symbols and Pictographs", aka emoji.