To make sense has several meanings.
A person, or any agent which is capable of some sort of reasoning, can make sense of some information, situation, or observation of the world. Secondly, some information, a decision, an action or a situation or arrangement can make sense, and this has a few meanings.
To make sense of something is to understand or accept it in some way, which can be intellectual or emotional. I'm finally making sense of chapter 3 in this Linear Algebra textbook. I still cannot make sense out of the past year's traumatic events in my personal life. Spirituality of some form helps many people make sense of their lives.
When a speaker says that some situation, information, and such makes sense, it can be a pronunciation of judgment that the situation or information is acceptable, agreeable or essentially good in some way. It might mean that it holds together logically, or that the speaker agrees with it based on his or her personal values, or on some emotional level. If a decision makes sense, it is considered well made, and if an action makes sense, the speaker approves of that action. Gun control makes sense to many people. Saving money for retirement makes sense. Programs to develop nuclear weapons do not make any sense.
When a speaker says that some information makes sense, it can simply mean that the speaker understands it: Chapter 3 of this Linear Algebra textbook is finally starting to make sense! (I am starting to understand chapter 3.) The text message you sent late Friday night doesn't make any sense. Were you drunk? (I cannot understand your text message.)
Make sense is also used to remark upon the connection between facts or observation, often when such a connection is newly discovered, but not necessarily so. Ah, it makes sense that he walks with a limp; I just remembered that someone told me he was wounded in the war. The opposite of this expresses the idea that two or more facts or observations appear inconsistent. It makes no sense that he lives in such a house and drives such a car on such a puny salary. He must have some hidden source of income.