There's no way to do justice to "Why is mathematics about real numbers?" within the length constraints of a Math.SE post, but here are some relatively philosophical observations and opinions (meant to be a bit provocative, in the spirit of answering a soft question).
First, as multiple people have commented, the real numbers are not universally regarded as the be-all/end-all number system. In The Road to Reality, for example, Penrose argues that complex numbers are more fundamental for physics.
Setting that aside, why do we count, and where did natural numbers, integers, and rational numbers come from? I'm not a historian, so everything below should be regarded as a parable, biased by modern mathematical training.
Counting (both the possibility and the ability) arises from the tension between variation and uniformity in the natural world:
Thanks to variation, there are "different types of thing": that piece of granite, that oak tree over there, the pine tree next to it.... If we look closely at the natural world, we find it to be made of unique objects, to occur in unique, irreproduceable events. In fact, the notion of "event" is our way of cutting the solid stream of existence into temporal and spatial chunks. As Heraclitus said, you cannot step in the same river twice.
Thanks to uniformity, there are recognizable "classes of things": rocks, trees, snowflakes, stars, sunrises.... No two rocks (or trees, or snowflakes...) are exactly alike. At the very least, they're "in different places" or "at different times" (otherwise they'd be identical).
Once the natural world is observed to contain classes of things, "counting" is a reasonable way to measure "how many/how much". In (a paraphrase of) Kronecker's famous quotation, The integers alone were created by God. All else [in mathematics] is the work of Man. To the contrary, the natural numbers (and therefore the integers) were created by us, as well, an abstraction for enumerating distinct objects similar enough to group together for some purpose.
To make a long story short:
An integer is a measure of additive comparison between two natural numbers. That is, it's a thing comprising a relationship between two other things. The standard construction of the integers in set theory merely formalizes this: An integer is an equivalence class of ordered pairs $(m_{1}, n_{1})$ of natural numbers, with $(m_{1}, n_{1}) \sim (m_{2}, n_{2})$ if and only if $m_{1} + n_{2} = m_{2} + n_{1}$.
A rational number is a measure of multiplicative comparison between two non-zero integers. The standard construction of the rationals blah, blah, blah.
It's unsurprising that both abstractions were invented: If two people have flocks of sheep (say), it's natural to ask "who has more sheep, and by how many?". It's natural to represent debts as negative integers. If a flock of sheep (well...) must be divided among several people, it's natural to ask "What is each person's portion?", and to use rational numbers to represent the answer.
The real numbers obviously arose many centuries later, under pressures of Archimedes' method of exhaustion (for which one needs numbers representing "limits of rational sequences"), and were formalized two millennia after that in order to put calculus on a solid logical footing.
I won't even touch the complex numbers, partly for last of time and space (heh), but mostly because Penrose (and many others) do so far more competently, with the depth the subject deserves.