Dear Akhil,

I am not an expert in simplicial methods by any means, but I thought it might help to give an answer at a much lower level than the other answers and comments.  What will come just reflects my own (somewhat meager!) attempts to understand some simplicial constructions.  My point here will *not* primarily be to explain why simplicial constructions beat cubical or other constructions, but just to give some examples of how you can use them and what they mean.  (Also, this answer is not very "high concept"; rather it is very very low concept!
But hopefully it might still be useful.)

Firstly, you can think of a simplicial set as just a big bag of simplices with instrutions on how to glue them: you have a set of points, a set of intervals, a set of $2$-simplices, etc., and the boundary maps tell you how to glue.  If you actually glue them according to the boundary maps, you get a space.  So at first blush it is reasonable to think of simplicial sets as just a technical improvement on the pretty simple idea of simplicial complexes.  Since reasonable spaces (from the point of view of algebraic topology, algebraic geometry, or smooth manifolds) can be triangulated, it is then not so surprising that one can capture a lot about topological spaces in this way.

Now let's suppose you have something a little more sophisticated instead, like a simplicial scheme: now you have a scheme of points, a scheme of 1-simplices, etc.

You can think of the scheme of points as just the basic scheme underlying the simplicial scheme; call it $X_0$.  Now the scheme $X_1$ of $1$-simplices has boundary maps to the scheme of points.  So you can think of $X_1$ as a kind of correspondence on $X_0$.  For simplicity, imagine that the two boundary maps into $X_0$ are closed embeddings, so that you have two copies of $X_1$ sitting inside $X_0$.   The fact that this is the scheme of $1$-simplices tells you that you are supposd to join all matching points in the two copies of $X_1$ by 1-simplices, and that you should think of these 1-simplices as varying continuously along the
two copies of $X_1$.  Now you glue in a family of 2-simplices indexed by $X_2$ in the same way, etc.

How do these arise: well a good example (taken from Deligne's *Hodge III* paper) is given
by considering the resolution of singularities $\tilde{X}$ of a singular projective variety $X$.  You can make the simplicial scheme $X_n:= \tilde{X}\times_X  \cdots \times_X \tilde{X}$ ($n+1$ copies) with boundary maps given by projections and degeneracies given by
partial diagonals.  (Side note: This construction does show one advantage of simplicial
constructions over various alternatives, namely, you can produce simplicial objects simply by taking iterated products; this provides a very convenient bridge between practice and theory, which might well be harder in other --- say cubical --- models.)


In particular $X_0$ is just $\tilde{X}$, so this simplicial scheme is $\tilde{X}$ with
a bunch of simplices attached.  If you think about how they are attached, you'll see that
the $1$-simplices join all the points that lie in a single fibre under the projection
$\tilde{X} \to X$.   And then every triangle made up of $1$-simplices bounds a $2$-simplex, and so on.

So this simplicial scheme is a model for $X$ in which the parameterizing schemes are smooth (edit: as Bhargav notes in a comment below, to actually get smooth schemes beyond $X_0$, one typically has to do more, but let me suppress this here), but one has glued in $1$-simplices explaining how points should be identified in order to get back down to $\tilde{X}$ (and the higher simplices are added just to ensure that no extra topological strucure is being created by the $1$-simplices you have glued in).

This indicates that working simplicially, one has flexibility in making certian constructions, e.g. rather than forming a quotient directly (like actually
passing from $\tilde{X}$ to $X$), we can instead form the quotient by gluing in paths between the points that are to be identified (and then adding higher order simplices as needed to kill of the loops, etc., that are accidentally introduced in the process of adding these paths).  

Of course, one can then go further to make constructions that would not actually be possible in the non-simplicial world.  E.g. suppose that you want to define relative etale cohomlogy $H^i(X,Y; \mathbb Q_{\ell})$, for a closed subscheme $Y$ of $X$.  Topologically, this
is the same as the (reduced) cohomology of the space obtained from $X$ by collapsing 
$Y$ to a point.  You can't usually do this in the world of schemes, but you can do it simplicially, using some variant of the construction described above.

So, to get an answer to the question "why does "simplicial" make everything work so well?", I would suggest that you not only think about the formalism (model categories and so on), but also that you play around with various constructions of the type I've described, and related ones (e.g. the constructions of $BG$ and $EG$ for an algebraic group as simplicial schemes), and try to picture them physically as schemes with simplices being glued in.  Try to think of other constructions from topology and see if you can figure out how you would make them in the world of schemes using simplicial schemes.  Of course, your explicit constructions will match with the general formalism, but they should also help to illuminate it, and to provide intuition.