By using the energy of sunlight, plants can convert carbon dioxide and water into carbohydrates and oxygen in a process called photosynthesis. As photosynthesis requires sunlight, this process only happens during the day. We often like to think of this as plants `breathing in carbon dioxide and `breathing out oxygen. However, the process is not exactly this simple. Just like animals, plants need to break down carbohydrates into energy. Oxygen is required to do this. Then why do the plants get rid of all the oxygen they produce during photosynthesis? The answer is, they do not. Plants actually hold on to a small amount of the oxygen they produced in photosynthesis and use that oxygen to break down carbohydrates to give them energy.

But what happens at night when there is no sunlight which is needed in photosynthesis? Interestingly, in order to maintain their metabolism and continue respiration at night, plants must absorb oxygen from the air and give off carbon dioxide (which is exactly what animals do). Fortunately for all of us oxygen breathers, plants produce approximately ten times more oxygen during the day that what they consume at night.

Plants break down sugar to energy using the same processes that we do. Oxygen is needed to break the sugar into carbon dioxide, releasing energy the plants can use to stay alive.

However, plants also take in energy from the sun(light), carbon dioxide from the atmosphere, and water from the soil; they use all of them in order to make sugar, and release oxygen. (They use the 'carbon' in carbon dioxide to build the sugar molecule). Since there's no sunlight at night, this gives the plants a way to stay alive, even when there's no light.

However, plants use sugar to build pretty much everything! Cellulose, the hard stuff in plants, is just a bunch of sugar molecules linked together. We can't digest it though, but some animals can. Similarly, plants make starch (sugar linked together, but not as tightly) to store energy for when it's dark. We're able to digest starch.

Since the plants use the sugar they make for more than just energy, they produce more oxygen than they use.

Great question! Plants produce oxygen, because when they photosynthesize, they take carbon dioxide (CO₂; a gas-form of carbon bonded to two oxygen molecules) and water (H₂O; an oxygen bonded to two hydrogen atoms) and combine them using light energy to produce sugars and oxygen. This stores the energy in chemical bonds (in the sugars) and releases O₂. The chemical equation for this is:
6CO₂ + 6H₂ C₆H₁₂O₆(sugar) + 6O₂

The plants use those sugars like we do when we consume them, for energy. Plants use the sugars they make by oxidizing them (with O₂, just like us) to release the energy stored in the bonds. They release CO₂ (just like us, when we breathe). But, when plants are photosynthesizing, they release more O₂ during photosynthesis than they will consume in respiration (oxidizing the sugars they have made). They release the oxygen through the same pores that allow the CO₂ to enter their leaf cells.

The quick answer to your question is that oxygen is just a waste product when plants do photosynthesis.

Plants can do two important things:
Use energy from the sun to turn CO₂ (carbon dioxide) and H₂O (water) into sugar (C₆H₁₂O₆) with oxygen (O₂) left over. This is photosynthesis.

And they can:
Break down the sugar (C₆H₁₂O₆) into CO₂ (carbon dioxide) and H₂O (water), but they need (O₂) oxygen to do it. This is cellular respiration.

We can only do the second thing.
The first law of thermodynamics tells us that matter cannot be created or destroyed. It cannot come from nothing and it cannot disappear. So the same number of atoms (C, H, O) have to enter and leave. Let us write photosynthesis as a balanced equation.

Photosynthesis:
6CO₂ + 6H₂O gives C₆H₁₂O₆ + 6O₂
Count up the number of carbon atoms on each side of the arrow. If you have six on one side, you need six on the other. Now count the hydrogen atoms. (6 X 2) on one side and 12 on the other. How many oxygen atoms are on the left side?
(6 X 2) + (6 X 1) = ___. Now how many oxygen atoms are in the glucose? 6.
So you have oxygen atoms left over. That is where the O₂ comes from. It is the left over material from making sugar. Just like when you make something, the scraps you cut off do not disappear. The plant breathes out the oxygen, which is good for all of us animals because we need oxygen, as you know.

Could there be animals without plants? Could there be plants without animals?

Plants produce oxygen as a waste product of making sugar using sunlight, carbon dioxide, and water. If a plant needs energy, but doesn't have sunlight, then it can burn the sugar that it made back when it had sunlight, and doing so requires oxygen.