Fine bouche is literally meaning: Fine mouth or Delicate mouth. It’s an expression that relates to the mouth you use to eat and the fact that people have to be delicate in their food choice: you are a picky eater, hence not everything will go through this mouth. You have this the expression like this.

“Fine bouche” means “thin mouth” (the prononciation is close in french and in english, by the way). It was originally “petite bouche” (“small mouth”). If you’re a gourmet, you’re more likely to eat your food little by little, not opening your mouth too wide and less likely to swallow big chunk without masticating. Thus the expression.

Like Papa Poule said in the comment, you should be careful with this expression : it imply that what was served was not good enough for you. The chef or host could ressent you. You could use the litteral translation for “I’m a picky-eater” (“Je suis un mangeur exigeant/difficile/pointilleux“) for added politeness.

Fine in this case is linked to the quality of the meal you are expecting. You can find epiceries fines or boucherie fine still in some places. There, you’ll find high quality or rare products.

Faire la fine bouche

Means been difficult and needing special treatment and better meals

Avoir un palais fin

Means having good and precise taste.

It’s also linked to un travail fin which means that it’s a work of high quality and high precision.