I actually have this as a new year's resolution. I'll try to summarize what's really worked for me so far.
Conversation is a complicated skill. Think of it like a sport. You'll improve slowly if you only train it in full matches with people. But you can buff up your training fast by picking a few key points to improve, and train them where there's no cost to failure (i.e. in private). In basketball/soccer, you practice your dribbles and shots first, before you stumble on them on the field.
The problem for me seems to be muscle memory. We freeze over certain things and going over certain routines. Some of us freeze when we're expected to respond, some of us freeze when trying to find facts. I used to freeze when trying to move a fact from memory to verbal speech.
So as training, I pick out an interesting topic. Just go with something on this forums. Like
pre-ordering in the games industry. Pick a side, for or against the topic. Doesn't matter, you're trying to practice arguing.
Write down a summary of this topic. Your summary should be in the form of
- Two main points that approve your side
- Support and evidences for those two main points
- Introduction and conclusion
Don't write out a whole speech. Just get your main points down there, e.g. "Pre-orders allow people to get a good price for something they're going to buy anyway" and "Pre-ordering lets a company gauge how much marketing support the game currently has". Make sure you're not writing out a full speech - the idea here is to latch on some main points and explain them naturally.
Try to write up your summary in just 5 minutes. If you're successful at doing this in 2 minutes or less, pick a harder topic, like explaining the role of a philosopher in a dwarven fortress. This is to train your brain to think up points for any topic quickly and then get those thoughts out through your mouth quickly.
Now look at the mirror. Imagine you looking at some nerd who you disagree with and want to explain your points. Talk, in a regular conversational volume. Volume is important, so you learn to avoid mumbling and hiding when you mess up or freeze under cognitive load.
You'll probably freeze and trail out a lot the first few times. Just practice explaining from one point of view on a topic until you stop freezing and get more comfortable with it. Then play through other topics whenever you have time to practice. The mirror helps a lot, because it trains you to face someone, even if it is yourself. I find myself standing up straight and using hand gestures better because of the mirror.
Once you're comfortable with that, try recording yourself and watching yourself again. You'll probably wince a bit, but you'll see where you're making mistakes.