One application I know is Scott's construction of forcing extensions of models of higher order theories of the Real numbers. Scott quickly after the invention of forcing, used a forcing argument to show that the Continuum Hypothesis is independent of an axiomatization of Second Order Analysis.
He interpreted first order variables (usually real numbers) as Random Variables over a complete Boolean Algebra with the ccc, second order variables (usually functions from the reals to the reals) as functions from RV to RV, satisfying a certain technical condition and then defined the Boolean value of each statement, which is an element of the Boolean algebra. He then showed that the axioms have Boolean Value $\mathbb{1}$ and that the inference rules respect the Boolean value (i.e. do not decrease the Boolean value) and finally exhibits a Boolean algebra in which the statement $CH$ has Boolean value $\ne \mathbb{1}$. See here for more details:
http://link.springer.com/content/pdf/10.1007%2FBF01705520.pdf
All in all this line of argumentation preshadows the way (set theoretic) forcing is presented in (for example) Jech's book, which explains forcing as forcing with Boolean valued models of the universe $V$.
My co-authors and I introduced a notion of dimension for forcing
extensions in the following paper:
Specifically, for any forcing extension $V\subset V[G]$, we defined
the essential size of the extension to be the smallest
cardinality in $V$ of a complete Boolean algebra $\mathbb{B}\in V$
such that $V[G]$ is realized as a forcing extension of $V$ using
$\mathbb{B}$, so that $V[G]=V[H]$ for some $V$-generic
$H\subset\mathbb{B}$. More recently, I have been inclined to call
this the forcing dimension of $V[G]$ over $V$.
This can indeed be seen as a dimension, in light of the following (essentially lemma 23 of the paper above):
Theorem. If $V\subset V[G]$ has essential size $\delta$, then
the essential size of any further extension $V[G][H]$ over $V$ is
at least $\delta$.
Proof. By combining the forcing into an iteration, we may view
$V[G][H]$ as a single-step forcing extension of $V$, and so it has
some essential size over $V$. Since we have an intermediate model
$V\subset V[G]\subset V[G][H]$, it follows by the intermediate
model theorem that $V[G]$ can be realized as an extension by a
complete subalgebra of that forcing notion. So the smallest size of
a complete Boolean algebra realizing $V[G]$ is not larger than the
smallest size of a compete Boolean algebra realizing $V[G][H]$ over
$V$. $\Box$
We had used the fact that there is a definable dimension, in the
forcing extensions over $L$, to show in general circumstances that
the modal logic of $\Gamma$-forcing over $L$ is contained in S4.3,
for a wide collection of forcing classes $\Gamma$.
Monroe Eskew points out in the comments that, contrary to my
initial thoughts about this, the size of the smallest partial order
giving rise to the extension will also serve as a forcing
dimension. The reason is that the density of a complete subalgebra of a complete Boolean algebra is at most the density of the whole algebra, simply by projecting any dense set of the larger algebra to the subalgebra. It follows by the same argument as in the theorem above that the poset-based forcing dimension of any intermediate model in a forcing extension is bounded by the minimal size of a partial order giving rise to the whole extension.
I propose that we officially adopt the poset-based notion as the forcing dimension of a forcing extension $V\subset V[G]$, denoting this dimension by $\left[V[G]\mathrel{:}\strut V\right]$. We may now observe the following attractive identity, confirming the suggestion of Will Brian.
Theorem. For any successive forcing extensions $V\subset
V[G]\subset V[G][H]$, we have
$$\left[V[G][H]\mathrel{:}\strut V\right]=\left[V[G]\mathrel{:}\strut V\right]\cdot\left[V[G][H]\mathrel{:}\strut V[G]\right].$$
Proof. Suppose that $G\subset\mathbb{P}\in V$ and
$H\subset\mathbb{Q}\in V[G]$, where these are the minimal-size partial
orders realizing the extensions. Suppose that $\mathbb{Q}$ has size
$\kappa$ in $V[G]$. So without loss there is $\mathbb{P}$-name for a relation $\dot\leq$
on $\kappa$, such that
$\mathbb{Q}=\langle\kappa,{\dot\leq}_G\rangle$ in $V[G]$. We can
now use the partial order $\{(p,\check\alpha)\mid
p\in\mathbb{P},\alpha<\kappa\}$, which is dense inside
$\mathbb{P}*\dot{\mathbb{Q}}$, to realize $V\subset V[G][H]$. This shows
$\leq$ of the desired identity.
Conversely, if we can realize $V[G][H]$ as a forcing extension of
$V$ by some partial order $\mathbb{R}$, then $V[G]$ arises as a
subforcing notion, and $V[G][H]\supset V[G]$ arises as quotient forcing.
The quotient forcing $\mathbb{R}/G$ can be thought of as the
conditions in $\mathbb{R}$ that are compatible with every element
of (the image of) $G$ in (the Boolean completion of) $\mathbb{R}$.
So the smallest partial order giving rise to $V[G][H]$ over $V[G]$
is at most the size of the smallest partial order giving rise to
$V[G][H]$ over $V$, as desired. $\Box$
Best Answer
The other answers are excellent, but let me augment them by offering an intuitive explanation of the kind you seem to seek.
In most forcing arguments, the main idea is to construct a partial order out of conditions that each consist of a tiny part of the generic object that we would like to add; each condition should make a tiny promise about what the new generic object will be like. The generic filter in effect provides a way to bundle these promises together into a coherent whole having the desired properties.
For example, with the Cohen forcing $\text{Add}(\omega,\theta)$, we want to add $\theta$ many new subsets of $\omega$, in order to violate CH, say. So we use conditions that specify finitely many bits in a $\omega\times\theta$ matrix of zeros and ones. Each condition makes a finite promise about how the entire matrix will be completed. The union of all conditions in the generic filter is a complete filling-in of the matrix. Genericity guarantees that each column of this matrix is a new real not present in the ground model and different from all other columns, since any finite condition can be extended so as to disagree on any particular column with any particular real or from any particular other column.
With the collapse forcing $\text{Coll}(\omega,\kappa)$, we want to add a new surjective function $f:\omega\to\kappa$. So we use conditions consisting of the finite partial functions $p:\omega\to\kappa$, ordered by extension. Each such condition is a tiny piece of the generic function we want to add, describing finitely much of it. The union of the generic filter provides a total function $g:\omega\to \kappa$, and the genericity of the filter will guarantee that $g$ is surjective, since for any $\alpha<\kappa$, any condition $p$ can be extended to a stronger condition having $\alpha$ in the range.
And similarly with many other forcing arguments. We design the partial order to consist of tiny pieces of the object that we are trying to add, each of which makes a small promise about the generic object. If $G$ is a generic filter for this partial order, then the union of $G$ is the joint collection of all these promises.
In many forcing arguments, it is not enough just to build a partial order consisting of tiny pieces of the desired object, since one also wants to know that the forcing preserves other features. For example, we want to know that the forcing does not inadvertently collapse cardinals or that it can be iterated without collapsing cardinals. This adds a wrinkle to the idea above, since one wants to use tiny pieces of the generic object, but impose other requirements on the conditions that will ensure that the partial order has a nice chain-condition or is proper and so on. So the design of a forcing notion is often a trade-off between these requirements---one must find a balance between simple-mindedly added pieces of the desired generic object and ensuring that the partial order has sufficient nice properties that it doesn't destroy too much.
In this sense, I would say that the difficult part of most forcing arguments is not the mastery of the forcing technology, the construction of the generic filter and of the model---although that aspect of forcing is indeed nontrivial---but rather it is the detailed design of the partial order to achieve the desired effect.