It doesn't necessarily mean during conversations (which can be and sound weird), but could simply mean when greeting them.
As an example, do you have a normal bar, restaurant, or coffee shop that you go to? When you go in, I'm sure that they greet you as most places do, and you probably toss of a "Hi" or a "Hey" and that's about it, even if you know their name. Try this next time. When you walk in and they greet you, give them a big smile like you're happy to see a friend and actually address them by name with a "Hey Mark! Can I get a Miller Lite" or "Hey Deb, table for four tonight". It doesn't really take much effort, but it really builds a connection.
WARNING: I'll toss this in as a warning. Don't fake knowing their name if you don't. A sincere "Hi" with a smile is still good. An "Oh hey...(looks at name tag) Jill..." comes off as fake. However, if they go ahead and give you their name ("Hi, I'm Jill and I'll be your waitress this evening") then by all means go ahead and say "Hi Jill" and address her by name throughout the evening.
It's worth reading, even if just to understand why others are recommending it. Sometimes they're not really recommending the book, but just "this book was the first to get me thinking about people skills".
The section on names is Chapter 3, ending with the principle "Remember that a person’s name is to that person the sweetest and most important sound in any language."
The chapter itself rambles on a bit and claims the success of US presidents & Andrew Carnegie is their ability to memorize thousands of first names... I think that's an exaggeration. But there's an element of truth to it, if you keep it in the back of your mind and try it yourself, memorizing people's names & using their name when you next meet them. (Especially at nightclubs, which is where I probably learned/used it most.)
Think of it as the difference between a letter addressed "Dear Ozovehe" vs "To Whom It May Concern". The latter shows they haven't even tried to get to know you personally. And now you'll notice when email / internet marketers try to use your name for just that reason....
As an example, do you have a normal bar, restaurant, or coffee shop that you go to? When you go in, I'm sure that they greet you as most places do, and you probably toss of a "Hi" or a "Hey" and that's about it, even if you know their name. Try this next time. When you walk in and they greet you, give them a big smile like you're happy to see a friend and actually address them by name with a "Hey Mark! Can I get a Miller Lite" or "Hey Deb, table for four tonight". It doesn't really take much effort, but it really builds a connection.
WARNING: I'll toss this in as a warning. Don't fake knowing their name if you don't. A sincere "Hi" with a smile is still good. An "Oh hey...(looks at name tag) Jill..." comes off as fake. However, if they go ahead and give you their name ("Hi, I'm Jill and I'll be your waitress this evening") then by all means go ahead and say "Hi Jill" and address her by name throughout the evening.