Just dig up the root or the part of the root that you want to use. You will find it difficult to remove it all so it should continue to grow in the same spot, but if necessary replant a bit of the root and it will soon grow again.
If you are planning to make horseradish sauce there are lots of recipes on line if you Google 'horseradish sauce'.
Just a warning that the fresh root is very powerful stuff and if you are planning to peel and grate it, keep it under water for peeling and don't rub your eyes!
There is nothing better than fresh horseradish sauce though.

Thank you Bertiefox Very useful information