Oxford Dictionaries, rest (verb), definition 1.4: (Of a problem or subject) be left without further investigation or discussion. Example: The council has urged the planning committee not to allow the matter to rest. That is the literal meaning of "rest". It has nothing to to with relaxation. The definition for "rest one's case" in the same Dictionaries is "Conclude one’s presentation of evidence and arguments in a lawsuit".

It means that the lawyer has finished presenting evidence or arguments.