Well, pretty much what it says on the tin.
I'm really curious about how pages like Statigram do their search functionality without users authentication and not exceeding the limits?
If I'm correct, Instagram API allows 5000 calls per hour, so I believe it's very likely that they indeed have more traffic than 5000 requests per hour.
Maybe It's a dumb question and Statigram has a special deal with Instagram to use their API or maybe they don't use the API and they use some other method?
Most likely they're using one of the following methods:
In cases like this, if you don't have a special arrangement, you're almost certainly violating the terms of service. If you think your service is useful enough that Instagram would be willing to whitelist you to make more requests, get in touch with them.