My question essentially breaks down to



> 
> How do you, a working mathematician, think about (real) symplectic groups? How do you visualize symplectic (linear) transformations? What intuition do you have?
> 
> 
> 


For example, we can think of rotations by thinking, "Okay, let's just restrict to a plane fixed by the transformation and see what it does there." This doesn't quite work for the symplectic group, though, since on the plane it is indistinguishable from the area-preserving linear transformations. The next smallest example is for a 4-dimensional vector space, which isn't particularly easy for me to visualize.



Intuition for symplectic groups
In a different direction from Victor Protsak's answer, I will focus on your comment



> 
> The next smallest example is for a 4-dimensional vector space, which
>  isn't particularly easy for me to visualize.
> 
> 
> 


It's true that the definition of the Lie group $\mathrm{Sp}(4,\mathbb{R})$ involves a $4$-dimensional vector space, but actually there are two nice $3$-dimensional homogeneous spaces on which this group acts.


The first, most obvious one, is the projective space $\mathbb{RP}^3$ of $1$-dimensional subspaces in $\mathbb{R}^4$.You can picture it as a compactification of $\mathbb{R}^3$ where you add a point at infinity for each possible direction of a line. The symplectic group acts on this space because it acts linearly on $\mathbb{R}^4$. The difference between the symplectic group and the full group of projective automorphisms $\mathrm{PGL}(4,\mathbb{R})$ is that $\mathrm{Sp}(4,\mathbb{R})$ does not act transitively on lines. Some lines are special, because they come from projectivizing Lagrangian planes. Conversely, any continuous transformation of $\mathbb{RP}^3$ which preserves lines and Lagrangian lines is an element of $\mathrm{Sp}(4,\mathbb{R})$ (this is a version of the fundamental theorem of projective geometry).


The second is the Lagrangian Grassmannian, the subset of the Grassmannian of $2$-planes consisting of planes where the symplectic form vanishes. It is a $3$-dimensional space again, which is a bit more topologically complicated. It's homeomorphic to $(\mathbb{S}^2\times \mathbb{S}^1)/\langle\iota\rangle$, where $\iota$ acts as the antipodal map on both spaces. This space admits an invariant conformal Lorentzian metric. This means that at each point (Lagrangian) $L$, there is a light cone, and in this case it consists of all Lagrangians intersecting $L$ in a line. Again, any continuous transformation of this space which preserves light cones comes from something in $\mathrm{Sp}(4,\mathbb{R})$. The Lorentzian structure is an accident coming from the isomorphism $\mathrm{Psp}(4,\mathbb{R})\cong \mathrm{SO}^0(3,2)$ and so does not exist for higher symplectic groups, but the incidence structure of Lagrangians is still present. You can imagine this space of Lagrangians as a compactification of $\mathbb{R}^{2,1}$, the $(2+1)$-dimensional Minkowski space, preserving its light cone structure.(see for instance <https://arxiv.org/pdf/0706.3055.pdf> for details)


If you get familiar with one or both of these spaces, it should help your intuition of how this particular group works.


**Edit**: Here is an additional description to answer R. van Dobben de Bruyn's comment


There is actually a more intuitive description due to Sophus Lie, the "Lie quadric". The space of Lagrangians in $\mathbb{R}^4$ is identified with the space of oriented circles on the $2$-sphere, where zero-radius circles ("point circles") are also allowed.


The way to set it up is to use $\mathbb{C}^2$ as your real 4-dimensional vector space, fix a complex-valued symplectic form $\omega$ and use $Im(\omega)$ as your real-valued symplectic form. Then, the projection $(\mathbb{C}^2\backslash\{0\})\rightarrow\mathbb{CP}^1$ sends Lagrangian planes to circles 2-to-1 (and so Lagrangians to oriented circles 1-to-1). Two Lagrangians intersect if and only if the corresponding circles are tangent. The group $\mathrm{Sp}(4,\mathbb{R})$ acts on oriented circles and points as tangency-preserving transformations. These transformations take a bit of time to get used to, since they can send a point to a circle... You just have to remember that, unlike in projective geometry, the transformations on the "space of circles" do not preserve points.


The transformations which do preserve the "point circles" are the same as the Moebius transformations acting on $\mathbb{CP}^1$. This is a $6$-dimensional subgroup of the $10$-dimensional $\mathrm{Sp}(4,\mathbb{R})$. There is also a "increase radius by $r$" operation which takes every circle in $\mathbb{C}\subset\mathbb{CP}^1$ to a circle centered at the same point, of radius $r$ bigger. Circles with negative orientation are thought of as having "negative radius". 
If two circles are externally tangent, they have opposite orientations,
 and if you add the same amount of "signed radius" to them, they stay tangent.


These two types of transformations (Moebius and radius addition) together generate the full group of tangency-preserving transformations. (for more details see <https://en.wikipedia.org/wiki/Lie_sphere_geometry>)


