IN THE COMMENTS: There's some questioning of the importance of the skill of speed listening. I add this:
I had some law school classes where I made recordings because there was more going on than I could pay enough attention to. I'd want to think about one thing, but I'd worry that I'd miss the next, which made it hard to understand anything properly. It's not necessarily the speed of the talking but the concision and complexity of the ideas.
You can tell when you're teaching that you can say something perfectly clearly -- your own recording will prove that to you -- and still know that students just can't absorb it in real time -- their questions will prove that. Often, you have to say the same thing three times before it's heard. If there were a recording, and the student could pause it and think, they'd understand it. I think it's helpful to speak a sentence fairly quickly (so the listener's mind doesn't wander) but then pause to let it sink in.
The subject of speed listening came up when we were playing Trivial Pursuit (90s edition), and we wanted to play faster so we tried reading the questions fast. The questions are written in a somewhat odd style, compressed, and studded with unexpected names. Reading them fast just slowed us down. We were all: Whaaaat?
One way to be disarming in conversation is to say one short but unusually phrased sentence with some surprising words in it and then stop. It forces everyone to try to understand what you've said. (There's some danger that they'll decide you're weird and ignore you.) It's much more effective than overexplaining everything and holding the floor too long.
I do think it's possible to train yourself to listen more efficiently. A lot of the problem is just the anxiety of worrying that you're going to miss something. It's most noticeable when someone gives you long driving directions or a phone number and you're not writing it down. It isn't really that hard to remember, but you're so worried that you'll forget that it becomes hard. Also, there should be a way to remember what was said and hold off until later to think through what it means. We all have the experience of replaying a conversation and realizing things about it that we couldn't figure out in real time. Sometimes it takes years to understand. The obstacle to speed listening, then, is that you are trying to combine listening and contemplation.
I think with a class, making a recording is usually unnecessary. Just take notes and write things that are truly perplexing down verbatim. Put a star or a question mark next to the things that confuse you and go back to those things very soon afterwards. If you can't make sense of them, ask outside of class or talk to other people about them. But don't wait too long to ask the teacher. Sometimes, I've had a student read me something they wrote in their notes and ask me what I meant when they wrote that. But it's a month after I said it!