I think that I understand why your teacher said this. We use "let" to mean "permit" , and "make" to mean "compel" . This can be through an action or by verbal means. For example: "When my dog wants to go outside, so I let him into the garden." (I open the door) "My dog doesn't want to go outside when it is raining, but I make him." (I push him outside) "On weekends I let my children stay up late, but if they have school the next day I make them go to bed early." So, if your teacher decided what you would study then they made you do it. But if the teacher said you could choose what to study then they let you do it.

He might be right cause to let someone to o something they just want to take a permision from you and make is like when someone makes you do something you have to do that cause that is like an obligation.It even perhaps the way he asked.

The best way to say this phrase is: “My teacher taught me grammar.”