Kogirei means neat, shipshape, clean, etc.
Danran is often used to describe a family, that is happy.
So "kogirei dewaaru ga danran no nai katei" means the home is well-kept, nicely clean but it lacks in family hapiness.
The speaker compares that home to a cage.
~という思い : a feeling/thought that is.....
~という檻 : a cage that is.....
To iu here indicate that its preceding clause explains the noun that follows.
The speaker has a feeling that he is chained in a cage which is a home which is nice and clean but without happiness. and the feeling is growing/mounting inside himself.
≪【こぎれいではあるがだんらんのない家庭】という【檻】の中に 、つながれている≫という≪思い≫がつのった。