Think like a wise man, but communicate in the language of the people. (William Butler Yeats)

This advice applies primarily to papers, but also to lectures and seminars (though one should also bear in mind that talks are not the same as papers).

On the one hand, one of the most important things in mathematics (though certainly not the only thing) is to get results, and prove them correctly. However, one also needs to make a good faith effort to communicate these results to their intended audience. (It may feel like you have attained some level of intellectual achievement if you can discuss a topic which is so difficult or jargon-heavy that most of your audience do not understand what you are talking about, but it is in fact a far greater intellectual achievement if you can actually communicate that difficult topic effectively to such an audience.)

Good exposition is hard work – almost as hard as good research, sometimes – and one may feel that having proved the result, one has no further obligation to explain it. However, this type of attitude tends to needlessly infuriate the very people who would otherwise be the strongest supporters and developers of your work, and is ultimately counter-productive. Thus, one should devote serious thought (and effort) to issues such as logical layout of a paper, choice and placement of notation, and the addition of heuristic, informal, motivational or overview material in the introduction and in other sections of a paper.

Ideally, at every point in the paper, the reader should know what the immediate goal is, what the long-term goal is, where various key statements or steps will be justified, why the notation, lemmas, and other material just introduced will be relevant to these goals, and have a reasonable idea of the context in which these arguments are placed in. (In short, a good paper should tell the reader “Why” and “Where” and not just “How” and “What”.)

In practice one tends to fall far short of such ideals, but there are often still ways one can make one’s papers more accessible without compromising the results. It sometimes helps to sit on a paper for a while, until the details have faded somewhat from your memory, and then reread it with a fresher perspective (and one closer to that of your typical audience); this can often highlight some significant issues with the exposition (e.g. use of some specialised jargon, without ever defining the term or citing a reference for it) which can then be easily addressed.

See also my advice on writing and submitting papers.