While we cannot identify the plant from its roots alone, these are likely roots from a nearby large shrub or tree. Tree roots in particular stretch well beyond their canopy edges (50 or more feet is commonplace for a mature shade tree) and will travel fastest to where they are finding the most suitable conditions - loose soil, moisture, and nutrients. Just like with trimming stems on shrubs, trimming roots (in this case, by cutting through them with a shovel) also can stimulate regrowth and branching, resulting in more growth than was there originally. This is why efforts to remove them may ironically only be resulting in more roots to have to deal with.

The only way to discourage them from entering the vegetable plot is to try blocking them with landscape fabric (also marketed as weed-barrier fabric). You can either dig-out the bottom of this bed and lay it onto the base and up the sides (as if it was a liner for a pond) before filling it back in again, or - and this may be easier - lay it on the ground and build a raised bed on top.

You could consider having a soil test done too, in order to see if there are particular nutrients lacking so you know which to supplement. (Supplementing all of them may not only be unnecessary or could stress the plants, but it can also save you money by not adding what you don't need.) Below is also the link to a page that discusses vegetable fertilizing so you can see which particular crops benefit from particular nutrient levels.

https://extension.umd.edu/resource/soil-testing-and-soil-testing-labs

https://extension.umd.edu/resource/fertilizing-vegetables

Miri