I am a German citizen. My husband has a UK residence card. Can he travel to France without a visa when he travels with me?
If the residence card says "residence card of a family member of a Union citizen" then yes, your husband may travel anywhere in the EU or the Schengen area without a visa when he travels with you (or to join you).
For a time, the UK issued residence cards that said "Residence card of a family member of an EEA national" instead of "... Union citizen." Those cards should also be accepted for visa-free travel, but the text does not comply with the directive, so it would be possible for a problem to arise.
If the card doesn't contain either text, then your husband should get a visa. However, if you are traveling by train or by ferry, you might be able to get past passport control with documentary proof of your marriage. There have been some reports here of this happening.
If your husband requires a visa, it should be free of charge and issued according to an accelerated procedure.