Leeches are small Annelids(segmented worms) , most species of which live in or near water. While most leeches are predators, eating small organisms whole, the most well-known leeches feed on blood or similar body fluids.
Medicinal leeches have been used since ancient India to treat various diseases(with little, if any, success). However, surgeons have found that leeches can help with circulation issues, skin grafts and transplants.
One of the many treatments Eustace had to endure during his nasty cold was a pair of leeches. When Courage called from inside the computer, the leeches inquired if the phone call was for them, and declared that they were too busy to take it.
It's unknown what Muriel was trying to accomplish with the two leeches, as leeches are not used for treating colds(and in fact, there really is no treatment for a cold).