I use 'appeal' to mean a request. However, it's not just any request, it's a request for someone to change their opinion or stance on something, often right before or after a major decision. It can often involves begging or underhanded behavior, but doesn't have to.
You would appeal a verdict in a court, you might appeal being sent on time-out, you might even appeal to a bully's better nature in trying to get them to back down.
You would probably not appeal someone's decision to go to the movies, or appeal someone's decision on which fruit and vegetables to buy.
In this circumstance, the sentence claims that sentimentality describes a wheedling, begging, underhanded attempt to 'win' an argument, debate, or other disagreement by evoking an emotional response, instead of reasoning out the right answer.
Let us imagine an illustrative scenario in which two people are arguing over, in a loss of airplane cabin pressure, whose oxygen mask one should put on first. Person A says that you should put on your own mask first, because then you're more able to help those around you. Person B says "HOW CAN YOU BE SO SELFISH! THINK OF THE CHILDREN! THEY GET A MASK FIRST! WHAT HEARTLESS MOSNTER WOULD PUT ON THEIR OWN MASK FIRST?!"
Person B's argument would be an example of sentimentality, as described in the quoted line provided by the OP. Obviously, their reasoning, if they had any, is wrong. Instead, they're trying to trigger an emotional response, to appeal to emotions, to win the point.