Yes, it's grammatical.
Here is how to understand it.
Let's assume that the average juror speaks and thinks in a certain way.
Here are the three clauses in your original sentence:
1) The average juror speaks and thinks that way.
2) They know (that) the average juror speaks and thinks that way.
3) They echo that way.
How can we combine 2) and 3) into a single sentence?
We can do that by joining the two clauses with that.
When we do this, that way in the main clause becomes the way that [they know ...],
and the other that way in the subordinate clause must be deleted.
4) They echo the way that they know the average juror speaks and thinks that way.
Note that this that is usually omitted. What's left is your sentence:
They echo the way they know the average juror speaks and thinks.