This required local implementation by each Member State. However, in more recent times EU food law has been in the form of regulations, making it directly applicable across all Member States but with a framework relating to local Member State administration and enforcement.
What will the UK do next?
It is likely that the UK will need to legislate in a number of areas of food law including labelling (or food information), food improvement agents, general food law, food packaging and nutrition and health claims. How the UK approaches this will be interesting as the focus in the UK across all areas of business in recent times has been de-regulation (The Red Tape Challenge). It may be that the UK implements more codes of practice rather than specific statutory rules in these areas going forward.