The sort of dishcloths generically known as J Cloths are made from a material called Viscose rayon:
This material is derived from cellulose and like cellulose it interacts with water. Water breaks hydrogen bonds formed within the fibres. This makes the fibres softer, and the exposed hydroxyl groups make the surface more hydrophilic. It's the latter process that makes a damp cloth more able to soak up water than a dry cloth.
The absorption of water on a fabric procedes by wicking. This requires a low contact angle and no hydrophobic areas on the fibres where the meniscus can get pinned. Incidentally, pinning is the basis of many superhydrophobes - even a relatively hydrophilic surface can be made hydrophobic by giving it the correct microstructure.
Anyhow, the contact angle of dry Viscose rayon is around 30-40º, which is fairly low but still high enough to prevent wicking and cause pinning. That's why a dry cloth is slow at absorbing water. It will absorb the water eventually, but the timescale may be many seconds or even minutes. After the Viscose has interacted with water and formed free hydroxyl groups at the surface the contact angle falls to effectively zero. This makes wicking, and therefore water absorption, much faster.
The dry cloth finds itself in a catch 22 situation. It has to interact with water to become hydrophilic, but until it becomes hydrophilic the water can't spread into the fabric for the interaction to occur. As anyone used to doing the washing up can tell you, the solution is to force the water to wet the fabric by wetting then squeezing it.