Why does my lower back tighten up when I walk or stand?

Back pain. Often as people age, their low backs become deconditioned: the muscles on the front of the spine too weak, and the muscles on the back of the spine too tight. Strengthening the front muscles and stretching the back muscles slowly, daily, can help. There can be other reasons also, involving the vertebrae and nerve roots, but these are less common. Consider seeing a physiatrist or neurologist.