Horses can, indeed, catch the flu. The symptoms are pretty much the same as flu in humans - coughing, a runny nose, and a high temperature. (If horses get body aches with the flu, they can't tell us, but it's likely).
Equine influenza is highly contagious, but almost never jumps into humans (It's far less likely to than avian or swine flu). It did, however, cause an epidemic in the American south a few years back when a strain of equine influenza jumped the species barrier into dogs.
Most working horses in modern times are vaccinated against influenza. The vaccine is administered yearly and, unlike in humans, is normally given in the early spring. This is because the high risk period for horses is not the winter but the active show season when horses from different barns mingle. Racehorses, which are at particularly high risk, are normally vaccinated every six months.
Treatment is about the same as for humans - the animal should be isolated from other horses and allowed to rest. If fever is bad, medication may be given to bring the horse's temperature down. Horses with respiratory symptoms may also be given cough syrup (yes, there is horse cough syrup). A more traditional remedy that might be seen in a medieval barn is "steaming" a horse. This process involves grabbing a couple of handfuls of hay and putting them in a water bucket, then half filling the bucket with boiling water. You then hold the horse's nose above the bucket - this clears out the sinuses. I was taught this method as recently as the 1980s and would still use it if I had a horse with a runny nose. (It's also effective on humans).
Mares with respiratory illness are likely to miscarry if early in pregnancy.