I'm Miss Minutes,and it's my job to catch you up before you stand trial for your crimes. 我是分钟小姐,我的工作就是在你接受罪行审判前让你了解情况。
catch you up
除了常见的身体的赶上你之外还有引申为为你脑补当前状况，知识等，让你跟上思维节奏的意思。意为让你明白，让你了解。
下面的意思也是异曲同工
What does that mean? - You'll catch up.
什么意思？你会了解的。
https://hinative.com/zh-CN/questions/244625
It means that you will enlighten the person on the details of something.
So by saying "I'll catch you up on the lecture from yesterday."
You're saying you will tell them what they might have missed or what they need to know.
Basically it means you'd teach the person.