The photos were square, and there were filters. Those features made things look cute.
Also getting a photo from your phone camera lens to the internet was often a multi-step process at that time. Instagram was two clicks and on your feed.
And the sharing model (all photos pushed to every follower) made people try a little harder on the images. That made it popular among amateur photographers. The profile model (grid of photos) also encouraged some level of vanity.
People discovered it by being invited by friends who like cataloging pretty things.
There simply wasn't a way to quickly share photos from your phone. Previously the best option was to mobile upload to facebook which was pretty common. Instagram made that process easier and also added the ability to quickly add "retro" looking filters/frames. It quickly became a way to see what your network was up to at that instant.
