You may be thinking that learning to bake cookies and learning something like chemistry aren't the same at all, and, in a way, you are right. But, the information you get from assessing what you have learned is the same. Brian used what he learned from each batch of cookies to improve the next batch. You learn from every homework assignment you complete and every quiz you take what you still need to study in order to know the material.
Another really good way to understand the importance of assessment is to think about learning skills. When playing basketball, for example, you get immediate feedback about how well you are doing, and this tells you how to adjust to get the ball in the hoop next time. When you are learning a skill, feedback (assessment) is automatic. When you are learning chemistry, the feedback process needs to be made visible through assessment.
Assessment doesn't have to be a written exam. You can determine if you have successfully learned something in a number of different ways, depending on what you are trying to learn. Recognizing that there are many different ways to assess learning and becoming skillful at self-assessment are important lifelong learning skills.