Given that the fixed end broke, with the clip still in place, I think it's fair to say it's nothing you did.

Without seeing the link itself, and/or the drivetrain it's impossible to say why it broke. It could be a fluke, defective link, or something specific to your bike like bad chainline.

However, just guessing based on what I see all to often, it might be that you're running your chain too tight. Disregard if you already know, but the proper chain tension is no tension for single speed bikes. The chain should always have some working slack, to allow for eccentricity in the sprockets. Ideal is vestigial slack with about 1/4" vertical free play in the center of the lower loop, at the tightest place.