I don’t understand exactly how to use this word in sentence and what does it really mean (constituent)
constituent= component (beeing a part of the whole)
Here: Czechoslovakia was divided into two countries which were before two constituent parts of one country.
Hi ! =))
You are right, a more natural to way to render the same idea is either “the integral part of the country”, or “the constituent of the country”, as the word “constituent” by itself implies the meaning of an “integral part!” 
Notably, you will come across this “constituent part of the country” ONLY on the Internet domains that are non-native English!