how was Sparta a war civilization?

Yes the Spartinian child raising (agoge) started directly after childbirth. An elder would inspect the child and throw it in a ravine near Sparta if it was seen unfit.

If the child was seen fit enough he would be summoned for the next stage when he was 7

He was send to a Spartinian school were he was taught the arts of war and was physically trained.

Spartan society was based around war. Early on the Spartans enslaved their neighbors in Laconia and Messenia so that all the menial tasks were done by slave labor while Spartan citizens focused on being supreme warriors. To become a Spartan citizen children went through a decade long training called agoge that taught them how to be the best fighters. To fail in this training meant to die. Compare this to Athens at the time where newly enlisted citizens only had two year military training (only one of which was actual combat intensive).