"Congress shall enact no law regarding the establishment of a religion".
The constitution does not define exactly what 'establishment' means. However, in the UK, the only country with an established church nowadays, it is understood to mean that the 30 or so most senior bishops of the Church of England have seats in the upper house of Parliament, and the PM chooses the bishops. Presumably the authors of the constitution wished to prevent that concept from crossing the Atlantic. Why those parents imagined that singing Christmas carols amounts to establishing a religion, is hard to imagine.