There is a moral story in my mother tongue that a parrot got disappointed as she was waiting patiently to eat the cotton fruit which will never be possible.How cotton fibre comes out?

no it turns to insect

Yes cotton flower becomes a fruit like any other fruiting/ flowering plant.

click on the link below to see a cotton flower =

http://www1.istockphoto.com/file_thumbview_approve/2670186/2/istockphoto_2670186_cotton_flower_gossypium_barbad ense.jpg

http://pubs.caes.uga.edu/caespubs/pubcd/B1252-graphics/B1252-16.jpg

Later on , after fertilization , the flower turns in to a fruit and that fruit CONTAINS in side the seeds .

Cotton fibers are the cellulose outgrowths / hair on the seed surface.

Click on the links below to see fruit , seeds and cotton fibers ==
http://www1.istockphoto.com/file_thumbview_approve/3220448/2/istockphoto_3220448_cotton_fruit.jpg

http://www.botgard.ucla.edu/html/botanytextbooks/economicbotany/images/Gossypium/Gossypiumbole.jpg

http://www.botgard.ucla.edu/html/botanytextbooks/economicbotany/images/Gossypium/Gossypiumcapsule.jpg

http://www.ferdinando.org.uk/images/cotton%20seed%20and%20fibre.jpg

Cotton life cycle ==
http://newbern.cpclib.org/images/NBCCjpg/nb29.jpg