Once all the skittles have mostly dissolved in each bottle, it's time to filter out all the extra leftover ingredients that we don't want in our finished product.
The original method that we used for this was to run the liquid through a filter made of several layers of paper towels pushed down into a strainer. As several commenters pointed out, it is likely that coffee filter might be a better solution than the paper towels. We haven't tried this ourselves, but it sounds quite reasonable and we see no reason why you shouldn't try it.
It will take some time to get all the liquid through the filter -- be patient. The stuff left in the filter will be pretty gross. We couldn't think of a good use for it, so we chucked it in the garbage.