Capillaries are are so narrow they can only allow a single erythrocyte through at the same time. The pressure from your heart causes fluid leakage from the thin vessels. Lymph ducts/nodes help get this fluid back into your blood. Since the heart doesn't pump lymph, that helps move it through your body is actually your musculoskeletal system. That is to a certain degree true from your venous / return flow in your "normal" circulatory system, too. It's why your feet can swell if you're sitting down too much. I'd speculate that lymphatic flow efficiency has local and systemic elements that contribute to it (and so your leg muscles moving lymph help your overall lymphatic system health).
Lymph system helps clear waste / ISF from the brain. Overall lymphatic circulation improvement from increased activity would intuitively improve lymphatic clearance from the brain. That would be true if there was direct lymphatic clearance or some indirectly / osmotic gradient driven process (I don't know which, or neither, is the case).
You might find this general overview of the body's lymphatic system interesting:
Edit: Here is a more direct answer to your question:
1) Increased localized pressure from contracting muscle pushes lymph away through a vessel.
2) Reduced localized pressure from relaxing muscle allows new lymph to refill the evacuated vessel portion.
3) Lymphatic vessels have valves that keep all this flow moving in the same direction.
4) Given that the lymphatic system is connected to the central nervous system, increased muscular contraction could increase drainage of the brain.