One extremely useful trick for visualising a certain class of simple 4- and 6-dimensional spaces is the toric moment map picture.

(a) The basic example is a 2-sphere $\{x^2+y^2+z^2=1\}$, which you equip with a linear height function $(x,y,z)\mapsto z$. Now instead of drawing the sphere you draw its image (an interval). Under this map, the sphere is a family of circles being collapsed to points.

(b) The next basic example is $S^2\times S^2$, which maps to a square: away from the edges, the preimage of a point is a 2-torus; over the edges away from the corners the preimages are circles; over the corners the preimages are points. Over each edge, there is a sphere whose projection to that edge is the one we saw in (a). The diagonal sphere $\{(x,x)\ :\ x\in S^2\}$ (respectively antidiagonal sphere $\{(x,-x)\ :\ x\in S^2\subset\mathbf{R}^3\}$) map to the diagonal/antidiagonal in the square and intersect each torus fibre in the diagonal/antidiagonal circle.

(c) $\mathbf{CP}^2$ with homogeneous coordinates $[x:y:z]$ projects to a triangle $\{a+b\leq 1,\ a,b\geq 0\}$ via $[x:y:z]\mapsto(|x|^2/T,|y|^2/T)$, $T=|x|^2+|y|^2+|z|^2$. Over each edge there is a sphere: you usually think of the sphere over the hypotenuse as being ``at infinity'' ($z=0$). These spheres are complex lines. If you cut out the spheres living over edges, everything retracts down to the fibre over the barycentre (which is again a torus).

In general what you're drawing is the image of a symplectic $2n$-manifold $X$ with a Hamiltonian action of the $n$-dimensional torus $T$ (in these cases, $n=1,2$) under the map $X\to X/T$. This is always a convex polytope whose vertices and be $\mathbf{Z}$-linearly identified with the vertex of the positive orthant in $\mathbf{R}^n$ (the Delzant property). A six-manifold projects to a 3-d polytope: $\mathbf{CP}^3$ becomes a standard simplex, for example. Various natural operations like blow-up can be easily visualised (chopping off corners of polytopes); certain natural singularities can be understood (by allowing non-Delzant vertices), for example the small resolution and flop of a 3-fold node has a nice toric picture (see the picture near the end of [this blog post][1]). High degree algebraic curves can be visualised using their amoebas.

Even more generally (as others in this thread have said), high-dimensional spaces can be visualised by their projections to other, simpler spaces. The most interesting and important part of this information is the singularities of the projection maps. This is the moral of Morse theory, Cerf theory, Picard-Lefschetz theory and, in this instance, of toric geometry, where the singularities of the moment maps occur along the faces and edges of the image polytope and give you a rich collection of important submanifolds for free.

More philosophically, I would say the key in developing a geometric intuition is in learning to draw simplified, lower-dimensional and possibly misleading pictures, provided you understand exactly how misleading the pictures are. For example, in the above example the diagonal and antidiagonal are disjoint in $S^2 \times S^2$ but their images intersect in the square.

  [1]: http://jonathanevans27.wordpress.com/2013/01/27/kronheimers-argument-small-resolutions-and-dehn-twists/