Why plants give off oxygen during photosynthesis?

Yes. All living things, except maybe the extremophiles that live near volcanic vents in the bottom of the sea, require oxygen to live. Even green plants need oxygen.

Sure, the leaves release oxygen, but the roots must draw in oxygen from air pockets in the soil to live. You can find this out by over-watering a potted plant. Saturating the soil will make the plant wilt and act as though it is drying up even though the soil is soaking wet. The roots have drowned and have died, therefore the rest of the plant runs out of water just as it would if the soil dried completely out.

Every living thing breaks down carbohydrates to release energy. This process requires oxygen. Even plants that live submerged in water need oxygen and get it from leaves and stems that float on the water surface - even kelp in the ocean does this. The things that look like plants in the ocean that don't have above water parts aren't really plants, but are animals that are just rooted in place such as coral, sponges, etc.