When you blow air through your Bubble Snake bottle, you are creating hundreds of tiny bubbles. As the air wiggles through the cheesecloth or towel, bubbles are continuously being made. The bubbles attach to each other when they come out of the fabric.
This is due to the surface tension of water. Surface tension of water tends to make bubbles unstable and due to this force they don’t last very long. But on adding soap the surface tension is lowered but still the bubbles aren’t very stable due to the force that water exerts on it so to reduce the area in contact with the water surface they tend to stick together as then they have a lesser area in contact with water and thus they are more stable as the total force is reduced.
In addition to this the surface tension of the water molecules that constitute the bubble exert an attractive force on the water(attraction between water molecules) in other bubbles and this force is responsible for holding the bubble together.