To "meet in the middle" (also: meet mid-way, meet halfway) is, put simply, to compromise during negotiations of some sort.
The context can be political or social, but is often applied to business deals. Each side concedes something in order to reach an agreement that benefits both sides in some way, even if neither side is fully satisfied with the result.
Example: "Company X avoided a long labor dispute with the union representing its workers. The union's demands were very high, but both sides met in the middle and a deal was reached."