'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.

Ok, I've checked this episode (never watched The Friends before) All of them consider Alan retarded or something. So he makes Ross to feel good or better about himself, because he (Ross) becomes more confident in presense of Alan. What do you think?

But I think or feel this is more like for friendship yet more as a mirror or just as a personal feeling indirectly if I don't think you are for relationship or I think more like as a imagination you may assume as your freedom in you mind and by your heart I am uncertain about. Just face true yourself, that answer is you.

That's a psychological thing, check this http://www.wikihow.com/Feel-Good-About-Yourself