The key geometric meaning is that flat families are those families where the fibers vary "continuously".  This notion allows one to talk about limits of families of algebraic varieties, which is particularly important in the study of deformation theory/moduli problems.

For instance, if you have a flat family of projective varieties, then as Timo points out, the dimension of each fiber is the same.  But more is true: the Hilbert polynomial of each fiber is also the same.  This allows degeneration techniques.  For instance, you can take a flat degeneration of your variety, compute a property about the degeneration, and then lift this information to your original variety.

I think that the geometric meaning of flatness is best understood via simple examples.  Consider first 
$\text{Spec}(k[x,y,t]/(xy-t))\to \text{Spec}(k[t])$ via the natural map.  This is a flat family.  You can see this geometrically, as the fiber over t is a hyperbola when $t\ne 0$, and as $t$ approaches $0$, the hyperbola gets sharper and sharper and then it "breaks" into two lines when $t=0$.

Constrast this example with $\text{Spec}(k[x,y,t]/(txy-t))\to \text{Spec}(k[t])$.  This is not a flat family.  Here, when $t\ne 0$, the fiber is always the same hyperbola {xy-1=0}.  But, when $t=0$, the fiber is an entire copy of $\text{Spec}(k[x,y])$.  This pathological variation of the fibers is encoded by the fact that this is not a flat family.