The main thing to visualize is the Hopf fibration of $S^2$, its suspensions, and their various compositions.
Let $f \colon S^3 \to S^2$ be the Hopf fibration.
When you suspend $f$ to get $Sf \colon S^4 \to S^3$, you effectively embed a 2-sphere as the equator of a 3-sphere and extend the mapping in parallel to 2-spheres of longitudes. Thus away from the poles you still have circles as preimages (the preimage of each pole is of course itself).
You can see that $f$ and $Sf$ compose to give a map $S^4 \to S^2$. To get a sense of how this looks as a fibration, you can work backwards. First, the preimage of a point in $S^2$ under $f$ is a circle in $S^3$. As noted above, each pointwise preimage of this circle under the suspension $Sf$ is again generically a circle. When the different circles fit together cleanly, it looks like you get a torus fibration, where the tori twist and interlink in $S^4$ analogously to the circles in $S^3$ of theHopf fibration.