To get the Veteran medal with description of "Restart the game 15 times", while in the middle of any stage, click the pause icon at the upper-right and you now get some options, one of which is Restart. Use that Restart option. Repeat 15 times.
Very nonintuitive medal because yes you can play the entire game through, delete save file, play it through again, etc (one possible way of interpreting "Restart the game 15 times") or you could refresh the browser page and continue with your game and do that 15 times (another possible way of interpreting "Restart the game 15 times") and no matter how much you did either you'd just be frustrated at this medal!
Game is basically just one of those grind until you get upgrades to get a little further, repeat until you've completed the game and all medals.