Disclaimer: Although Instagram is indeed very popular and does an excellent job of adhering to solid design principles, it is important to point out Instagram isn't a one size fits all solution. It is more important to understand the problem you are solving including the context in which you are solving it in order to yield the best result for your users.
1. Context = Who + What + Where + When
Instagram understands the context in which it operates extremely well which is a large part of why it is so successful. People like documenting their life online. Taking pictures is a great way to do this since many people carry a camera phone everywhere they go. Instagram provides an easy way to record your life's photo journal using a smartphone.
The app went through many iterations until most of the major problems were solved...
- Easily sharing your pictures with family and friends (or even just yourself)
- Quickly applying filters to set the mood of your picture
- Having pictures backed up in the cloud in case you break or lose your phone
The decision to crop everything as a square wasn't an accident. Squares are a solid shape which easily scale on any device. Most pictures on Instagram are captured using a smartphone and by cropping all images to the width of the device leaves room for simple editing options near the bottom of the screen easily accessible by a thumb on a single hand.
2. Less is More ... Responsive
Card layouts work well on pretty much any device and were around long before Instagram. The thing Instagram does really well is not clutter the interface with stats and text. They show only the most critical piece of data -- the picture itself.
Stats can be seen by hovering over or clicking on a picture but the initial page load is just enough square image cards to fill the screen along with a short bio of who the images belong to.
Notice how all the user uploaded content are squares? This provides contrast to the other stuff on the page like the big circle LOAD MORE
button...