How to write a brief research statement for a postdoc application?
The best thing to do is to try finding examples of research statements written by people in similar positions. Many people put their research statement on their website. From these you can see the style and length that is typical in your field.

Your research statement should discuss both the work you have done and the work you want to do in the future. It should also discuss the bigger picture that links these together.

Generally, less is more. People instinctively write long research statements describing all their work in detail. This instinct must be fought, as no one will read a 10-page research statement. For a postdoc, I think 2-3 pages (not including bibliography) is typical, at least in my area of theoretical computer science. Faculty-level research statements will be a bit longer.

Keep your audience in mind. (I.e. does the person reading this know your area?) Proof read and polish meticulously. And definitely ask other people to look at your draft and give feedback.

Good luck!