This was puzzling me as well, when I first thought about it. It seems as if the real line remains at its original size all the time, then, when $t=1$, it suddenly implodes to a single point during an infinitely short period of time, and this appears somewhat discontinuous, although we assure ourselves that all the maps are continuous.
HEAVILY EDITED (last paragraph deals with difference to visual perception)
To understand intuitively that nothing evil is happening here, the following picture might be helpful:
Imagine the real line as a bar (of infinite length) which can be rotated in Euclidean plane around an axis going through zero. Let's say the real line is aligned horizontally. Formally a rotation of 90 degree counterclockwise during the unit interval at constant speed corresponds to a function $r$ mapping a pair $(x,t)$, where $x\in\mathbb R$ and $t\in I$, to the point $(x\cdot cos(\frac12\pi t),\ x\cdot sin(\frac12\pi t))$. The homotopy $f_t$ is then obtained by composing with the projection onto the $x$-axis, i.e. $f_t(x)=x\cdot cos(\frac12\pi t)$. Since this is just a rotation, it is continuous, but the image of the entire real line under the projection does not change continuously.
To see the difference between the mere projection onto the $x$-axis and what you would actually experience if you observed the rotation from under the $x$-axis: Think of yourself standing in a completely flat world, being able to see infinitely far in every direction. You'd start by looking at your shoes, seeing only the surface, then lifting your head, and eventually you will see the horizon, even though the surface of this imaginary world is infinitely vast and completely flat. Continue raising your head and the horizon will lower until it leaves your field of view. This would be what you'd see in the situation above from under the rotation axis. This shows how the projection onto the $x$-axis is different from the projection to your own eyes. If eyes were flat then you would only ever see an intervall of $\mathbb R$ for $t<1$ and nothing for $t=1$. Since the eye is curved, things are different, which is also the reason why object farther away appear smaller.
Hope this was helpful and my English was sufficient to explain what I mean :-)