We are able to display them on the web because Instagram has made their API available to the public. This means that Instagram allows third-party websites like us who offer Instagram users various services to access Instagram's basic (and public) data.
The data includes information like photos, comments, and likes, to name a few.
We can only display public information that you, as an Instagram user, have agreed to make public.
We do not store the data in our system, modify or permanently remove it.
(One way to understand API is by comparing it to a set of building blocks. When it is open to the public, it becomes accessible by the world, allowing other programmers to build upon it. What they build can offer users more options or an enhanced experience.)