Disc Analysis allows the player to analyze the Harmonics Disc to upgrade the character through learning new Skills, expanding Skill Slots, etc.
Once you select the point you want to analyze, you can check the Point Cost necessary for upgrading and the conditions for unlocking. Verse Maidens have strengths and weaknesses. The stronger she is in that area, the fewer points she'll need for analysis; the weaker she is, the more points she'll need.
Analyzing a point causes new points to be arrayed, so it is important to consider how one goes about analyzing. Also, among the analysis points are those that cannot be analyzed unless particular conditions are met.
The points needed for analysis can be obtained during Level Up and Proficiency Up. Since Skills can only be learned through Disc Analysis, it's imperative one doesn't forget to analyze then they level up EXP and Mic proficiency.