Why do units (from physics) behave like numbers? What are units (like meters $m$, seconds $s$, kilogram $kg$, …) from a mathematical point of view?
I've made the observation that units "behave like numbers". For example, we can divide them (as in $m/s$, which is a unit of speed), and also square them (the unit of acceleration is $\frac{m}{s^2}$). In addition to that, we can cancel units:
$$s = v\cdot t$$
If for example $v=4\frac{m}{s}$ and $t=5s$, then
$$\require{cancel}s=4\frac{m}{\cancel s}\cdot 5\cancel s=20m.$$
Note that $\frac{m}{s}$ can also be written as $ms^{-1}$. This is another example where units "behave" like numbers.
So why can we cancel units, why do units behave like numbers?
I want to get an answer that can be understood by highschool students.

This question is closely related to the influential essay
Wigner, Eugene P. The unreasonable effectiveness of mathematics in the natural sciences [Comm. Pure Appl. Math. 13 (1960), 1–14; Zbl 102, 7].  Mathematical analysis of physical systems,  1–14, Van Nostrand Reinhold, New York, 1985.
Historian Grattan-Guinness published a response to this in
Grattan-Guinness, Ivor.
Solving Wigner's mystery: the reasonable (though perhaps limited) effectiveness of mathematics in the natural sciences. 
Math. Intelligencer  30  (2008),  no. 3, 7–17. 
Grattan-Guinness's point is that the question is backwards.  The following is perhaps not exactly his point but is related.  When the Greeks or other ancient philosophers first delved into science and mathematics they typically dealt with magnitudes or quantities.  These were always tied to the physical entities being studied, whether objects being counted or frequencies arising in music theory.
Eventually magnitudes and quantities were abstracted into what we know today as natural numbers, which were ultimately extended to broader number systems such as the hyperreals. The point is that the effectiveness of numbers in describing reality stems from the fact that the numbers themselves have magnitudes and quantities as source. The puzzle, according to Grattan-Guinness, only arises if we forget their source.
Thus what the OP refers to as the "units" were the source of the numbers, rather than vice versa. A philosopher would therefore describe such a problem as a self-inflicted pseudoproblem.
