that gentleman to be the best dressed
"...the gentleman who is best dressed..."
Note that "gentleman" has two possible meanings. The first is a man with good manners and good moral behaviour; and the second is a man born into an upper-class family. Or it might mean both. It was usual for upper-class people to think that only upper-class people could have good manners and good moral behaviour.
"Best dressed" means "the person with the best clothes". What does "best" mean here? We'll find out!
whose dress no one observes.
"... has clothes which do not stand out in a way which people notice."
The clothes he is wearing are "normal" or "average". If his clothes were unusual in some way, people would remember him as "the man who wore that shirt". His clothes are similar to everyone else's though.
Why does this matter? We need to remember that this man is upper-class, so he is rich. He can probably afford very good clothes. With his rich friends, he can wear those expensive, high-quality clothes, because all his rich friends are wearing clothes of the same quality. Wearing smart clothes is also a sign of respect, so if you wear scruffy clothes when you meet your friends, you are being disrespectful to your friends.
What happens when he goes out around town though? And more than that, what happens if he goes somewhere where everyone is poor? They can't afford high-quality clothes. Maybe they're lucky if they even have shoes. Everyone else there is equally poor, so they don't feel bad compared to each other. But a rich man walking through the town in his fine clothes will remind them of how poor they really are, and that will make them feel bad - especially parents who can't afford that for their children. He is noticed because his clothes stand out, and it feels to the poorer people that he is wearing those fine clothes to say to them "I'm better than you are".
So... To be respectful to the people who cannot afford the same quality of clothes, he should wear clothes which are not such high quality. Perhaps not as low-quality as the poorer people are wearing, but still not as high quality as he would wear with his friends. Then his clothes don't stand out, and it doesn't feel like he's showing off. And as a result, he is "better dressed" - and a better gentleman.