How to substitute x into binomial 
Write down the first three terms in the binomial expansion of $(1 – x)^{15}$.
  By substituting $x = 0.01$, find an approximate value for $0.99^{15}$.

I understand the first part however how do I substitute $x$?
The first three values are:
$-x^{15} + 15x^{14} - 105x^{13}$.

You are starting at the wrong end of the binomial.  The first term should be $1^{15}$.  This order is chosen because $x$ is much smaller than $1$, so the terms will decrease as you go along.  To substitute $x=0.01$, just rewrite the term for that value.  So if the exponent were $4,$, the first terms would be $1^4-4x^3+6x^2$ and substituting in $x=0.01$ would give $1-0.04+0.0006=0.9606$.  The point is to see that the first few terms give a good approximation to the correct value, here $0.99^4=0.9659601$
