Attendance at school is mandatory for all children in Germany from the age of six until the age of eighteen, and home schooling is not permissible. Children often have a choice between public and private schools. The latter may be religious or secular, and either can obtain governmental subsidies if they are properly accredited. Aside from a few private universities, attendance at colleges and universities is free, and stipends and loans are provided to students who cannot defray their living expenses while studying. 
Nothing really ominous about it, in fact until recently many States in the United States didn't allow homeschooling. Back in 1980, home schooling was illegal in 30 states. It was not until 1993 that all 50 states made the practice lawful in the United States.
The Romeikes had several options. They could have sent their children to an accredited private religious school at government expense. They could have sent their children to an unaccredited private school at their own expense. They could have sent their children to a secular private or public school. Since Germany is part of the European Union they could have moved to another country in the EU that allowed homeschooling or they could work to change the educational laws in Germany so that homeschooling would be legal, while obeying the law. Germany is a Federal Republic and has a democratic government pretty much like ours.
Instead they choose to ignore all the LEGAL options for educating their children and BROKE THE LAW. Now HSLDA who paid for the Romeikes' to come to the United States and a bunch of Fundamentalist Christians are beating their chest and demanding that we allow these LAW BREAKING MOOCHERS to stay in the United States.
Let's be clear this is NOT about religious persecution. Germany has religious freedom. The German Constitution (much like our own) protects freedom of religion by guaranteeing free exercise of religion, banning the establishment of a state church, and providing some forms of affirmative governmental support to religious and ideological organizations.
This is about a family choosing to break their countries educational laws with the encouragement of HSLDA and being paid to flee to the United States and stir up controversy.