The refill tube does exactly what its name states— it refills the toilet. This part interacts closest with the overflow tube and the fill valve. As the fill valve is signaling for the tank to refill, a little bit of this incoming water flows from the fill valve, into the refill tube, and then into the overflow tube, which dumps clean water into the bowl. The refill tube is an integral part of the system, as it tells the overflow tube how much water to input into the bowl. Without a properly functioning refill tube, the toilet bowl will either have no water, or overflow with too much water.
The refill tube comes upwards in an arch from the fill valve, and then drains down directly into the flapper, which releases the clean water into the bowl. If the fill valve is not properly secured to the refill tube, (or the refill tube is not attached to the overflow) the water will not go back into the bowl, and will instead stay stuck in the tank. Most importantly, the refill tube must always stay above the overflow tube, otherwise the water will flow backwards and keep the tube full.