You don't need to remove anything except for space considerations, and what you have in there already is the best 'starter' there is. Leave picking out the worms and all that until much later.

My suggestion: add shredded, somewhat damp cardboard, or shredded newspaper, or cardboard down below and newspaper on top, to fill the bins up completely to top. (You didn't say how deep the bins are but I'm guessing they are not full). You can mix a small amount of the existing 'dirt' in with the cardboard. Leave it for a while. Start adding smallish amounts of food waste, ideally well chopped or even pureed - pull back some cardboard to create a hole, put the stuff in, lay the cardboard back over top. Watch to see if the worms that are in there sow up in the food waste you've left. When you see that most or all of the food waste is 'gone' or not identifiable, you can add more food to a different part of the bin.

Come back here later and let us know how it's going. Key tips: go very slow with adding food until things look and smell all right (not stinky). The cardboard / paper (called 'bedding') will also eventually shrink, settle and disappear, it's good to add more. Keep the bedding from drying out completely, but keep slightly damp (a bit on the dry side is better than soaking wet). It's fine if the top layer of bedding (esp shredded paper) is dry as long as there is moisture down below (the top layer will tend to dry out just from being exposed to air, it also buffers what is below).

You can remove some of the 'dirt' (actually worm castings) a bit at a time, and if there are plenty of worms elsewhere, it's not a big deal if some get removed at the same time - they'll reproduce. Usually worms will gravitate to the parts of the bin they like best (usually where the best food and bedding is); you can remove whatever else you want (e.g. castings) where there are less worms. But I'd probably wait before removing anything unless you really need the space.

You said they are 60 liter plastic containers, but didn't say what they look like (I know you said you tried to post a photo - try again on a reply! It works for me.) Depending on what shape and configuration the bins are in, there are various methods you can use to harvest some of the vermicompost for use. Generally speaking, it sounds like so far you have the basics down pretty well - add some veggie scraps from time to time, and add shredded newspaper or other bedding to absorb moisture.

You have a very good head start on most vermicompost newbies because you have established bins with a high population of worms that sounds like they are already functioning well. Contrary to what you might think based on other animals (and also contrary to what you might read from some vermicomposting experts), worms absolutely love living in an environment of their own slime and poop. The more they've gone over and through the material in a bin, the better they like it. I conclude that this is because of the ways in which their slime and poop feed a population of beneficial microorganisms that keep the environment just the way they like it, and prepare their food for them the way they like it.

So I would not encourage you to just dump everything out, pick the worms out, and start "fresh and clean" as this is the opposite of what worms like! Also, picking worms out of finished vermicompost can be a lot of fiddly effort. It is less labor intensive to clear out just a portion of the bin, refill that with damp shredded paper or other bedding material, mix in a little food, and let the worms gradually move out from their established environment in search of the new food. If you have a bin that is wider than it is deep, you can scoop out one half of it and do this, only add new food to the half with new bedding for a couple months or so, then next time scoop out the other half.

How soon do you want to start using the vermicompost, and what do you plan to use it for?

Benn - I cannot understand why you could not find numerous links to "beginners how-to" if you Googled "vermicomposting"! My search of the word just now returned 742,000 results...

Anyway, I have found this one quite useful: Red Worm Composting . I'd also suggest that you invest in Mary Appelhof's nice little reference book called Worms Eat My Garbage. Amazon has it for $11 new, but I saw it offered for as low as $1.99 from used vendors.