Kindergeld (child benefit)
Applying for Kindergeld as an EU citizen
If you are a national of an EU country, you apply for Kindergeld with the same application as everybody else.
The EU annex (KG 1-AnEU) is added to it. It asks where you work, where you are insured and whether something similar is paid in the other country.
You do not have to enclose a residence permit for this. The residence permit is the card or the sticker in your passport that permits you to stay in Germany; it is called for only from nationals outside the EU.
Why the EU annex is asked for
Within the EU it holds that only ever one country pays. The EU annex gathers the details from which it follows which country that is. You do not have to judge that yourself; you only enter where work is done and where people live.
When the child lives in another EU country
Then the foreign annex is added as well. Both sheets together are the normal case for families living across two countries.
What that costs
The price depends on how many children you are applying for Kindergeld for. The sample calculations are on the page about Kindergeld (child benefit).