Things generally tend to relax toward lower energy. In the water’s case, wetting the fabric via capillary action is a lower energy state than sitting in the bowl. Gravity is of course one of many forces acting on the water. If a force is larger than the gravity, the water has no problem going up. In a way, it’s “falling down” the energy potential of the fabric. It’s not as obvious as say, a magnet lifting a metallic object (of the right composition) or water moving up under pressure. It’s not one-way, so you might have also had water that had dripped down onto the counter.
The water in the wet fabric is also interacting with the atmosphere. In typical household conditions, the air is not at maximum humidity so water will also tend to evaporate. As water is removed from the fabric, more can be pulled from the bowl. One way of testing/demonstrating this would be having a bowl of water as the control, and different shapes of paper and fabric acting as wicks, propped up and hanging over the edge.
Candle wicks and oil lamps actually use this, though instead of just evaporating, the wax/oil goes into a gas phase and burns.
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