One could say many things about this, and I hope you get many replies! Here are some remarks, although much of this might already be familiar or obvious to you.
In some vague sense, the study of simplicial objects is "homotopical mathematics", while the study of objects is "ordinary mathematics". Here by "homotopical mathematics", I mean the philosophy that among other things say that whenever you have a set in ordinary mathematics, you should instead consider a space, with the property that taking pi_0 of this space recovers the original set. In particular, this should be done for Hom sets, so we should have Hom spaces instead. This is formalized in various frameworks, such as infinity-categories, simplicial model categories, and A-infinity categories. Here "space" can mean many different things, in these examples: infinity-category, simplicial set, or chain complex respectively.
For intuition, it helps to think of a simplicial object as an object with a topology. For example, a simplicial set is like a topological space, a simplical ring is like a topological ring etc. The precise statements usually takes the form of a Quillen equivalence of model categories between the simplicial objects and a suitable category of topological objects. Simplicial sets are Quillen equivalent to compactly generated topological spaces, and I think a similar statement holds if you replace sets by rings, although I am not sure if you need any hypotheses here.
If you like homological algebra, it helps to think of a simplical object as analogous to a chain complex. The precise statements are given by various generalizations of the Dold-Kan correspondence. For simplical rings, they should correspond to chain complexes with a product, more precisely DGAs. Again, one has to be a bit careful with the precise statements. I think the following is true: Simplical commutative unital k-algebras are Quillen equivalent to connective commutative differential graded k-algebras, provided k is a Q-algebra.
A remark about the word "simplical": A simplical object in a category C is a functor from the Delta category into C, but for almost all purposes the Delta category could be replaced with any test category in the sense of Grothendieck, see this nLab post for some discussion which doesn't use the terminology of test categories.
Since you used the tag "derived stuff" I guess you are already aware of Toen's derived stacks. Some of his articles have introductions which explain why one would like to use simplicial rings instead of rings. See in particular his really nice lecture notes from a course in Barcelona last year.
I tried to write a blog post on some of this a while ago, there might be something useful there, especially relating to motivation from algebraic geometry.