There are three things mentioned here in this passage - all three of which are wrong. The first two are examples, while the third is what the proverb is seeking to help us understand is wisdom. This proverb is one that I would call a "considerate" proverb. God is trying to get us to understand what being considerate means. The situation is that of someone who is troubled or sad about a situation or situations in life. The trouble has reached the point of troubling their heart. When this is the case, we need to understand that we need to be gracious and kind. A person who sings songs to someone who it troubled - is being inconsiderate toward the person. They will not think it a blessing to have happy songs sung to them. Even less would be the suggestion that they join in singing, which would then be presented at the perfect thing to help them get out of their emotional doldrums.
This kind of action is compared to someone who takes off their outer garment on a cold day. This would not be a blessing. The person would immediately react negatively to the cold and want to wrap up again. The other instance mentioned is someone who pours vinegar on soda. There would be an instand negative reaction as the soda vinegar mixture would begin bubbling up ruining both the soda and the vinegar. The purpose of giving these two examples if to remind us that as we would think either of them an unwise thing to do - so we should realize that someone who is dealing with a troubled heart should not do so by singing happy songs.
A wise man ministers to someone based on their need - not based on his own feelings at the time. A man is troubled not just because he is not feeling good emotionally. He is troubled because there are genuine issues that are unmet. He needs godly counsel. He needs someone who wants to understand him and the circumstances that have him so troubled. That kind of loving counsel will minister to him effectively.