How did the Darrys come to be such ardent Targaryen loyalists?

Well, the Darrys have been Targaryen loyalists for some time. They got a lot of Harroway lands from King Maegor, they backed the blacks in the Dance and suffered horrible losses, they fought for Daeron II (hence the business with the Clanking Dragon Inn), the Demon of Darry became Lord Commander of the Kingsguard, and in the reign of Aerys II Ser Willem Darry was made master-of-arms at the Red Keep and his brother Ser Jonothor Darry was made a member of the Kingsguard, and three Darry brothers including Ser Jonothor died at the Trident.

So it’s a combination of past rewards (remember the hiddden tapestries) and past sufferings.