'the way' means 'how' here.
I like the way she smiles - I like how she smiles.
I don't like the way she looks at me - I don't like how she looks at me.
this 'how' is not clear and can only be understood from the context.
In this dialogue, though, it's certain that this 'way' is a good one because the person likes it. However, we can't really know what this 'way' is. Perhaps, this persons feels more confident or more beautiful or smarter,etc.