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.
Stenosis. There are multiple potential causes, but I. Your age group, spinal stenosis is probably most common cause. In spinal stenosis, the spinal canal is narrowed and patients will difficulty standing or walking for prolonged periods. They will often feel the need to lean forward, and so they can be seen at the grocery store leaning on the cart as they walk.