I can't tell you much about the relation to Voevodsky's work, but I can give you a quick summary of Derived Algebraic Geometry.

In DAG you enlarge the category of geometric objects that you can study. It is easiest done by using the functorial approach. So a scheme is just a functor from commutative rings to sets.

Now we'll first enlarge this category in the stacky direction. This means that we change the target of our functor. Instead of studying only set-valued functors we'll allow groupoid-valued functors. This leads to the stacks you are used to. But for some complicated moduli problems groupoids can't encode enough information. So we'll make the target category even bigger and allow for simplicial set valued functors. So now you have to study the category of functors from commutative rings to simplicial sets, also called the category of simplicial presheaves. In this category you have to impose the right descent and atlas conditions to find the objects that have the right to be called geometric. These guys will be called higher stacks.

I think that what we have done so far is very similar to Voevodsky's construction. But I am absolutely not an expert on this. Probably one of the experts will show up here soon and explain that.

So far we haven't derived anything. That starts when you also enlarge the domain category. The natural "derived category" for commutative rings is simplicial commuative rings. That's because the category of commutative rings is not abelian, and then simplicial objects are a good replacement for chain complexes. Derived algebraic geometry then is the study of functors from simplicial commutative rings to simplicial sets, or simplicial presheaves on simplicial commutative rings. Again you have to find the functors inside this category with the right descent and atlas conditions, and that's a lot of work. These guys are then called derived schemes, derived stacks and derived higher stacks.

The geometric intuition for these derived schemes and stacks are that they are ordinary schemes plus a fuzzy cloud of nilpotents on steroids around them. They can encode much more information in their structure sheaf than ordinary schemes. A good example is the intersection of two subschemes in an ambient scheme. You can then construct a "derived intersection". This derived intersection intrinsically in its structure sheaf has encoded that it is an intersection, something you could never accomplish with normal nilpotents.

The difference between the Toen-Vezzosi approach and the approach of Lurie is that TV use model categories where Lurie uses infinity-1 categories. Which approach you actually use is a matter of taste. I think the analogy is to either work coordinate free or with coordinates.

A final comment: If you look into the papers on Luries website or into Toen-Vezzosis Homotopical Algebraic Geomtery II book, you'll find that they work in much greater generality. They do the whole program not on the category of simplicial commutative ring, but for quite general model categories. If you really are only interested in DAG, there are Toen's course notes on his homepage or Luries original thesis.