You've not joined any of the other tables, hence the results are only bringing back the tblAvailabilities rows.
var query = (from avail in dc.tblAvailabilitities
join aaTyp in dc.tblAvailabilityAppointmentType
on avail.availabilityID == aaType.availabilityID
join appType in dc.tblAppointmentType
on aaType.appTypeID == appType.appTypeID
select new { TypeName = appType.appTypeName });