The success rate is the number of times a puzzle was completed divided by the number of times it was started. So every time someone starts a puzzle and then abandons it (because they couldn't figure out the next step, they got distracted by something else, their computer crashed, or whatever) before coming up with the correct solution, it drives down the success rate.
I'm not sure exactly how the penalty rate is calculated, but it's probably the number of times a correct solution was reached after incurring a penalty (for submitting a wrong answer or whatever) divided by the total number of times a correct solution was reached. Of course, the developers my be using some other formula, like the total number of penalties incurred over the total number of times the puzzle was started. However they're calculating it, it's a measure of how often penalties are incurred on that puzzle, and everyone who solves the puzzle with no penalties will drive the rate down.
Neither number affects your scores, but they do give you an indication of how hard the puzzle might be.