Communicating mathematics
Solution 1:
As an experienced teacher, let me offer a few bits of advice.
By all means, do go to talks and see what makes some speakers effective (in general, perhaps not always to you) and what makes other speakers less so.
One of the arts of teaching is to anticipate where the audience will have difficulties. In large part, this comes with experience. But it's important to think about it when you're preparing your own lectures. If you've already seen that students have had issues understanding certain concepts or certain types of computations, try to address that head-on in your presentations.
Another mistake that many teachers make (sadly, even those with lots of experience) is to get mired in proofs/derivations before the students even have a context or any reason to care. Well-chosen examples done before the general theory help a lot. Indeed, I would say the same thing holds for you to understand a theorem: You really don't understand it until you can give me one or two examples (perhaps one of them trivial) to illustrate its significance and its application. This is part of the motivation to which others have referred. :)
Last, as far as teaching is concerned, letting the students know that you care about their learning and progress is huge. It's ok, once you establish rapport, to be honest about your disappointment in their test scores, etc., but they have to know that you want them to learn. And, Jeff, particularly in relatively low-level courses, they do not want to sit through your doing pedantic proofs that would satisfy you as a student. It's much more important for you to teach your precalculus and calculus students how to approach problems and how to write up understandable solutions of the problems. They don't need to know (let alone understand) the proof of the Maximum Value Theorem in Calc I; they do need to know how to give a solid application of it in the context of an applied max/min problem. They need to know how to set up functions in a logical way, think about their domains, understand why the function should have a maximum (other than just "the problem asked for it"), and then find it. (Many students these days have such weak algebra skills that those weak skills get in the way of so much more conceptually interesting stuff.)
And, last, I would concur with the folks on here encouraging you to answer more questions by way of learning to practice explaining things at different levels. Sometimes on here it's very hard to assess the level of the asker. I've noticed several graduate students (at least I think they are) giving explanations that would be appropriate for people at their level, rather than for the one who asked (who was clearly in an introductory undergraduate course).
OK, so my notion of "a few bits" is askew. :)
Solution 2:
Most people, when presented with a claim, need a certain amount of time and/or contextual information to understand not why that claim holds, but what it means and why it matters.
Simple example: if you were to present to me a theorem about simple groups, having defined them, I would be staring at you wondering why I should care, because I have no concept of what makes simple groups important.
Solution 3:
It is very hard to know when precision is going to trade off with clarity and when precision is going to be necessary for clarity. I don't pretend to know. However, two defaults:
at all levels of math, you should probably go a little slower than your instincts tell you to go. Do not worry about insulting an expert by going too slow. If the expert wants you to speed up, the expert will indicate that. Then you can speed up. Similarly, most students appreciate a slow clear explanation and are not insulted (in fact are happy) when you explain something they already know.
pausing to let the other person talk is extremely helpful. I find I have to force myself to do this.