Why does 1 cubic meter = 1000000 cubic centimeters?
Take a look at the Wikipedia article for SI prefixes, in particular, the following excerpt:

When units occur in exponentiation, for example, in square and cubic forms, the multiplication prefix must be considered part of the unit, and thus included in the exponentiation.

$1\ km^2$ means one square kilometre, or the area of a square of $1000\ m$ by $1000\ m$ and not 1000 square metres.

$2\ Mm^3$ means two cubic megametres, or the volume of two cubes of $1000000\ m$ by $1000000\ m$ by $1000000\ m$ or $2\times10^{18}\ m^3$, and not 2000000 cubic metres ($2\times10^{6}\ m^3$).



Thus, the relationship between the cubic metre and the cubic centimetre is as follows:
$$1\ m^3 = (1\ m)^3 = (100\ cm)^3 = 100\ cm \times 100\ cm \times 100\ cm =  1000000\ cm^3 = 10^6\ cm^3$$
$$1\ cm^3 = (1\ cm)^3 = (0.01\ m)^3 =  0.01\ m \times 0.01\ m \times 0.01\ m = 0.000001\ m^3 = 10^{-6}\ m^3$$
For a real case, converting $243.7\ cm^3$ to $m^3$ step by step:
$$243.7\ cm^3 = 243.7\times 1\ cm^3 = 243.7\times (1\ cm)^3 =243.7\times (0.01\ m)^3 = 243.7\times 0.01\ m \times 0.01\ m \times 0.01\ m = 243.7\times 0.000001\ m^3 = 243.7\times 10^{-6}\ m^3 = 2.437 \times 10^{-4}\ m^3$$
This is simply the way it was defined, and it is the standard everyone adheres to. More examples of SI prefix usage can be found on this page from one of the organizations which created the SI units, the BIPM.