cURL (Client URL Library) is a library for making HTTP requests. It's not anything specific to classes (nor does it actually use classes). It's often used, as in this case, to communicate with a remote server.

There are other, more simple ways of fetching data over HTTP from PHP. For example, you can use the file_get_contents function to read the response into a string.
cURL is used in this case because it supports things like cookies and file uploading which the API makes use of.