'makes sense' does not mean 'understand clearly'. You can say 'That makes sense to me.' but it still doesn't mean 'I understand it clearly.' It's not as emphatic. Usually 'is sensible' can replace 'makes sense' His advice makes sense. His advice is sensible. There is another usage - 'to not be able to make sense of something'. which is closer to 'understand' I can't make sense of what they're doing. I don't understand what they're doing.

Isolating a word that makes up an idiom sometimes might not make sense, as idioms most times might not be directly related to the words they’re made of.

It's hard to separate the meaning of "sense" within "to make sense" because as you say, it is an idiom. So it's easier to think of "make sense" as one verb. Sometimes "it makes sense" means that it seems like a good idea: > "It makes sense to carry an umbrella if it might rain." > "I asked him for advice, and everything he said made sense." > "Sleeping for eight hours a night makes a lot of sense." Sometimes "makes sense" means that it is understandable: > "Oh, now that you have explained it, it makes sense." > "You keep trying to explain when to use the past perfect in English, but it still doesn't make sense to me." > "That guy talks a lot but he makes no sense." > "It makes sense that you're going to Mexico if you want to learn Spanish."

The word "sense" has many different meanings in English. In the idiom "to make sense," it means something like "to be logical, reasonable or wise." You can see this meaning of the word in many uses and forms, e.g.: "I try to eat a sensible diet." (I have good, healthy reasons for what I choose to eat.) "You were talking nonsense in your sleep." (The words you were saying were not logical; they had no clear meaning.) "I'm sorry I kissed you. I lost my senses in all the excitement of the New Year." (I am claiming that my kiss was not for a good reason; my judgment was impaired by the emotion of a moment and perhaps by alcohol.) "This math problem makes no sense." (The problem may be logical, but it goes beyond my own logical ability to understand right now.)

Sometimes "make sense" is used to mean "what you are saying is logical" or "it's good advice". It makes sense to leave early. You skip the traffic.