Important information on what we currently know about how the Brexit decision effects EU students can be seen here via the UK Council for International Student Affairs (UKCISA).
As per the advice via UKCISA here you can still enter the UK freely and have an automatic right of residence for up to three months. After the initial 3 months you must be in the UK exercising the right to reside as a student. In order to have a right of residence in the UK as a student you have to have Comprehensive Sickness Insurance (CSI). Please view here for essential information about CSI. If you do not already have CSI we recommend that you take out an appropriate policy as soon as possible.
You do not have to register to stay in the UK. If you would like to do so you can apply for a registration certificate to confirm your right of residence as a student. If you have family members who are not themselves EEA or Swiss nationals, you may want to apply for a registration certificate as this can make it easier for your family to apply for an EEA family permit or residence card. Having a registration certificate may also make it easier to apply for permanent residence in the future should you wish to do so. If you did wish to apply for a registration certificate guidance on how to do this and the process can be seen here.
You can also view a video from Free Movement regarding an EU settlement application here and some updated information from UKCISA about the EU Settlement Scheme here.