From Rotman's Algebraic Topology:
I've been trying to come up with a simple example for this definition but I can't seem to understand in even a simple example how to visually interpret $F(u,t) = \phi(t)$.
Can someone give a simple example of a level homotopy that's easy to understand?
Note that $I^n/\partial I^n\cong S^{n-1}$, so any level homotopy $F:I^n\times I\rightarrow X$ gives rise to an unbased homotopy $f:S^{n-1}\times I\rightarrow X$ satisfying $f(\ast,t)=\varphi(t)$, so think of the level homotopy $F$ as defining family of $(n-1)$-spheres in $X$ parametrized along a curve $\varphi$. In the definition, however, we any not requiring $f_t$ to be injective, so the spheres may be degenerate.
In the simplest case consider $X=\mathbb{R}^3$ with the level homotopy given by a map $f:S^{1}\times I\rightarrow \mathbb{R}^3$. Then this is just a cylinder in $\mathbb{R}^3$, only it may twist about freely (and continuously) and it may self intersect or crush itself to a point. Explicit examples include the inclusion $S^1\times I\hookrightarrow \mathbb{R}^3$, or "twice" the inclusion, which traces the cylinder upwards once then back downwards along itself. There is a cone, where the radius of the cylinder shrinks (enlarges), or an hourglass where the cylinder pinches to a point in the middle