By the Federal Aviation Administration's account, Francis W. Tapler is one of 173 air traffic controllers at the Leesburg regional control center who went on strike illegally last Aug. 3 and have paid the price--dismissal.

Tapler tells a different story: He had just undergone nasal surgery and was absent on sick leave when the strike began. Suffering insomnia, cramps and "extreme nervousness," he lay at home on the following days, taking tranquilizers on doctor's orders and not totally aware of what was happening at the center.

Yesterday, the 35-year-old Tapler pleaded his case at a hearing of the Merit Systems Protection Board, the federal agency that represents the last hope for reinstatement short of the courts for most of 11,500 controllers fired after the nationwide strike.

In coming months, about 300 controllers fired from jobs in the Washington area are scheduled to follow Tapler and appear for hearings before members of the board's regional office in Fairfax County.

Federal courts have ruled that the strike was illegal, so controllers must make other arguments to explain their absence--that they were coerced into staying away from work, were sick or didn't know the strike was under way in time to meet the deadline for return to work, for instance. Proof that the FAA committed procedural errors in firing them, such as giving them insufficient time to answer the charges, could also reverse their dismissals.

Twenty-nine controllers were reinstated shortly after the strike began. In recent weeks, the FAA has reached private rehiring settlements with 21 more to avoid the board appeals, a Department of Transportation spokesman said.

But the cases of close to 10,000 other controllers remain before the board, more than doubling its normal caseload. So far, it has completed the exhaustive hearings process on only 250 cases, according to the spokesman, and has made decisions on 14. All but one firing was upheld, the spokesman said.

Yesterday, in a hearing room in Fairfax County, attorneys representing the FAA questioned Tapler closely about his health and conduct during the strike, as his own legal counsel looked on. Later, his father, his wife and a neighbor took the stand to support his claims that he had been seriously ill, was not a striker and was fired unfairly.

They were the equivalent of witnesses for the defense. On the other side, Angelo Viselli, chief of the Leesburg center, testified that supervisors repeatedly attempted to contact Tapler, a controller for 12 years, at his Leesburg home and verify his status, but that he had not responded and was assumed to be part of the strike by the Professional Air Traffic Controllers Organization.

Yesterday's hearing--there was argument, for instance, over why notations on a personnel log were in different color inks and whether it mattered--is typical of the detail the Fairfax office and the board's 10 other regional offices will have to sort out in clearing the backlog.

As with other cases, testimony was directed toward a "presiding official" who acts as a judge. Both Tapler and the government can appeal the official's eventual decision to the board itself. If he chooses, Tapler could appeal a full board ruling in federal court.

The board hopes to have all cases heard by Dec. 31. The process has been slow so far, however, because of legal maneuvering by both sides and budget restrictions that have prevented the board from hiring additional staff.

Meanwhile, as supervisors, military controllers and nonstriking civilians run the control system, the FAA is trying to train a new corps of controllers. An FAA spokesman said that so far 1,500 controllers have completed training and 1,600 more are taking the courses.

The air traffic system now handles about 83 percent of prestrike volumes, according to the FAA, and will be back to normal by July 1984. FAA projections have been criticized on Capitol Hill as optimistic.

The FAA continues to rule out a mass rehiring of the fired controllers. According to a PATCO spokesman, about 50 percent of them have now found full-time jobs. About 200 have got or seem likely to get controller jobs in foreign countries such as Australia and Papua New Guinea.

Speaking to reporters after his testimony, Tapler said that he had worked as a handyman in a Leesburg electronics shop for three months but was let go in February and has been unemployed since. His wife has begun working and they have dipped into $21,000 in retirement funds that the FAA paid out.

Tapler said his background has made it hard to find work. At job interviews, he said, "They say, 'We'd like to hire you,' " but then add "you'll probably" get your old job back as a controller.