I basically had the exact same problem with my garage door. I didn't want to replace it at the time so I fixed it. In my case it was all across the bottom and I basically custom made a new bottom for it, piece by piece. I'll try to take some pictures later and include here.
If the only rotten part is the thing shown in the photo, it is really not that bad and much, much simpler than my repairs. You can take that panel off, cut off the rotten part, and basically cut pieces of wood to fill it back in.
All that said, I must warn you that wood garage doors require constant maintenance, and the repairs that I mentioned take quite some time and patience. If time spent is a major concern, I can tell it won't be worth it. Personally, I just enjoyed the challenge.
Still, I suspect that fixing the garage door is only part of the problem. My garage door rotted because the gutters in front of the door constant fill and don't drain properly, causing the water to splash right in front of the garage door. I would try to make sure to resolve that issue too (and be constantly vigilant about) is that is your issue too.