When changing fields, should I focus more on describing past research or connecting it to the new area in my statement of purpose?
I am going to focus my answer on your original question statement (but without getting involved in editing the official title of your Question):

When changing fields, should my past research be written in a way that brings out the maximum relevance to the new area?
In my current draft of this part, about a half is about this the problem and the result of the work, another half is for my contribution and what I've learnt.

Good.  This is exactly what they're looking for, in that section of your essay.

Having said that, should I focus less on describing the work and leave space for the thing that will be useful for the next research?

No, you can let the &quot;previous work&quot; section stand alone.  That's the section where you show the committee that you have done good science, and that you can speak about it in an understandable way.
Just to make sure we're on the same page -- your essay is going to have another major section, where you get to be imaginative about what you would like to do in the new field, right?