How can I multiply by time? I'm reading this article about collision detection.
In it, he says:

However, t appears to be referring to time - The time of impact to the object?

How am I meant to be able to multiply by time? What even would "Time" be? Apart from the above screenshot, there is no other explanation as to what "Time" could be.
 A: The intuitive idea is we want to create a type of movie.  In the movie, we have a sphere and a (triangular) plane: the sphere starts at some point (which we call the base point), and moves with some constant velocity.  We let $t$ range from $0$ to $1$.  Time $0$ corresponding to the start of the movie, time $1$ corresponding to the end of the movie.  We'd like to know when (if ever) during the movie the sphere is intersecting the plane.
The observation in the second paragraph you posted is that, if the sphere sphere ever intersects the plane during our movie, then immediately before it intersects the plane, it will have to just touch the plane, and immediately after the sphere finishes intersecting the plane, it will just touch the plane on the opposite side.  Thus, if we can solve for the times $t_0$ when the sphere just touches the plane on the near side, and $t_1$ when the sphere just touches the plane on the far side, we know that the sphere intersects the plane at all times between $t_0$ and $t_1$.  Of course, one could imagine a situation where we couldn't find $t_0$ or $t_1$ (or where we couldn't find either): I'll leave it to you to think about what these situations mean, geometrically.
