Eyesome is an adjective that means pleasant to look at.
The suffix S-O-M-E means ‘characterized by being.’ So adding it to the word eye, in this case, means characterized by being pleasant to the eye. Getting imported into the English language, our word of the day is almost always meant to be flattering. Here’s an example:
When I captured the attention of everyone in the bar, I thought it was because of my eyesome appearance. It turned out they were all just staring at my mismatched shoes.