Hi Igor,
In the answer “I’ll see you two weeks later,” the word “later” really means “after” because we have put it with a period of time, “two weeks” in this case. So you’re really saying, “I’ll see you two weeks after.” But two weeks after what? This implies that I will see you two weeks after some point in the future. For example, “I’ll return from vacation next Monday and then I’ll see you two weeks later” would be a correct way to use this.
But the correct answer in your grammar book question is two weeks after right now (“Bye for now”), not two weeks after a time in the future. In this case, we use “in two weeks’ time.” This just means “two weeks into the future from right now.”
We may also use the phrase “See you later” because there isn’t a period of time used with “later.”
I hope that helps!