"How does sugar water affect bean plant growth?"
Putting sugar in the water will possibly have several affects on plants. One effect you have already seen by noticing that the soil stays moister in the pots watered with sugar water.
Water moves across a membrane by a process called osmosis.
When you add sugar to your water you are changing the osmotic potential of the pure
water. Less water will move into the root because of this change in osmotic potential so the soil will be moister. I believe this was the main question you wanted answered. One way that the sugar water may affect plant growth is that it could influence microorganism growth in the area
surrounding the roots. This may be good for the plants or bad for the
plants. The sugar concentration may also have an effect.
Maybe a little is good or a lot is bad. Only your experiment can show you the effects.
Sugar Water Effect Plants.