In case you need to grab follower count (or other fields) without logging in, Instagram is nice enough to put them in JSON inside the page source:
$raw = file_get_contents('https://www.instagram.com/USERNAME'); //replace with user
preg_match('/\"edge_followed_by\"\:\s?\{\"count\"\:\s?([0-9]+)/',$raw,$m);
print intval($m[1]);
//returns "123"
Hope that helps.
24 May 2016 Updated to be more tolerant of spaces in JSON.
19 Apr 2018 Updated to use new “edge_” prefix.