Plants need oxygen but don't have a heart
Complex animals all have a circulatory system of some sort to get oxygen throughout the body for respiration. Plants also do respiration even though they are net producers of oxygen through photosynthesis.
Plants also have a bit of a circulatory system, but for the most part it isn't nearly efficient enough to deliver oxygen everywhere in the plant: plants don't have anything like circulating hemoglobin-filled cells we are familiar with in vertebrates, and although fluids flow in plant circulatory systems, they aren't circulating quickly like you are familiar with in your own body. This means that plants need to get oxygen near to where it is needed.
Roots, in particular, consume oxygen
The problem of a plant not getting enough oxygen through its roots isn't because the whole plant gets oxygen through the roots, it's because the roots themselves need oxygen to function. Roots do a lot of 'heavy lifting' in a plant, literally: they are pumping ions across membranes to pull in water, to concentrate other nutrients that the plant needs for survival and growth, and to pressurize the plant enough for those nutrients to make it up into the leaves. Those processes take energy, and in turn they need oxygen.
Additionally, photosynthesis isn't taking place in the vicinity of the roots: plants are both taking in CO2 and releasing oxygen from their stomata in the leaves and stems.
If roots cannot get enough oxygen, for example due to overwatering, the roots themselves are damaged, and this harms the plant as a whole.
Because there are always exceptions in biology, it should be noted that some plants are indeed adapted to wet environments and have better systems for promoting the diffusion of oxygen including increasing the amount of air space in the roots, where oxygen diffuses more quickly than through aqueous media. Other plants use fermentation in their roots to survive temporary hypoxia.
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