It indeed appears some residue on the board is causing conduction.
It looks like there are no parts on the board (except the headers) so:
I would let the whole board soak in alcohol. Isopropal 91% would be OK but ethyl alcohol from home depot would be better.
Soak it for 5 minutes or so then brush it off with an old (stiff) toothbrush. Repeat a few times.
Follow this by a good washing in hot water and regular dishwashing soap (and the brush). Rinse in hot water then dry as best you can. An air hose would be good here.
Not sure of your flux so I'm not sure which of above will be the most effective, however there is no reason not to perform both.
For the future I would suggest you get a small soldering iron. The one I use has a 1/16" wide tip. You will also need some small solder. I would suggest 0.031" (0.8mm) no clean 63/37 tin lead solder. I do not recommend no lead solder for those just starting out. The lead bearing solder flows better and is less likely to produce a cold solder joint.
Good luck.