I wouldn't ask how to keep your dog from being fearful but how to better secure him in his contained area. I agree with 'my boy diesel', your dog instigated the situation by going into another's yard.
If it was my dog that got bit I would press for euthanizing the dog that started the issue. Especially if they're a neighbor and it could happen again in the future.
What you can do now is just be apologetic, pay for the bills, and hope for the best. Potentially having fearful dog should be the least of your worries.