开窍 (kāiqiào) is a common and versatile word used in informal and semi-formal conversations.
In Education and Learning: This is a classic context. It's used when a student, after much effort, finally understands a difficult subject or problem.
In Personal Relationships and Growth: It's often used to describe a person who was stubborn, immature, or naive, but has finally seen reason or matured. For example, someone who finally realizes their partner is bad for them has “开窍了”.
In Problem-Solving and Work: In a business or creative setting, it can describe the moment a team finds the solution to a tricky problem or when an artist overcomes a creative block.
Connotation: The connotation is generally positive, as a breakthrough is a good thing. However, as mentioned above, it can be used sarcastically or condescendingly to imply someone was slow on the uptake.