Well, there are several ways that higher dimensions get used. One is
through what are called "configuration spaces". (See Banchoff, Beyond
the Third Dimension, chapter 7). For example, if you are working with
a process for which you are recording four or more pieces of
information, then your data naturally lies in a higher-dimensional
space. For instance, a paleoecologist at Brown University studies the
size of a various tree populations on the border of a lake over time.
He does this by looking at the pollen concentrations for the different
trees as they are recorded in the yearly mud deposits at the bottom of
the lake. By taking core-samples at various locations within the
lake, he can determine the relative numbers of the different trees at
different locations on the shore of the lake.

How does this relate to higher dimensions? Well, for each tree, there
are four pieces of data: the (x,y) coordinates of the core-sample,
the concentration of pollen, and the year of the count (represented by
the depth of yearly deposit in the core sample). This data is
four-dimensional; moreover, the geometry of the four-dimensional data
also has meaning. For example, the level surface for a specific
concentration tells what time and place the specific concentration can
be found, while the level surface for a given time would be the graph
of the concentration over the area of the lake for that given time.

Another example is the "double pendulum": take one pendulum and hang
a second one off then end of it:

                        o
                       /
                     /
                   ()
                     \
                       \
                         \
                           ()

The dynamics of the double pendulum turns out to be very interesting,
and complex. It can be chaotic, and the pendula can swing over the
top and so on (depending on the relative weights, lengths and speeds
of motion).

Then the position of the weight for the first pendulum lies on a
circle around the top swivel, so it has coordinates (x1,y1) in
relation to the swivel. Similarly, the position of the lower weight
lies on a circle centered at the middle weight, so can be represented
by (x2,y2) lying on a circle. Thus the state of the system can be
represented by (x1,y1,x2,y2), a point in four dimensions. Moreover,
the pairs of coordinates lie on circles, so the set of points in 4D
that represent valid positions of the system form a torus (or
doughnut) in 4D. (This is because for every (x1,y1) on the circle,
there is a whole circle of positions for the other pendulum, so the
configuration space is a circle cross a circle, or a torus.)

Moreover, as the system changes over time, the configuration positions
change over time, so (x1(t),y1(t),x2(t),y2(t)) forms a PATH of points
lying on the torus in four-space. So if we want to understand the
dynamics of the system, we need to understand paths on a torus in
four-space. In this way, our geometric understanding of objects in 4D
help us to understand three-dimensional (or even two-dimensional as in
this case) phenomena. (Once we add time to the system above, we could
even consider the configuration to be 5-dimensional, with coordinates
(x1,y1,x2,y2,t).)

Hope this gives you something to think about.

Davide