Read cyclotomic polynomial first. Here's why such a polynomial has a sequence of coefficients which reads the same in both directions, maybe with a change of signs. Suppose t is a primitive n'th root of unity. Then so is its conjugate. But since |t|=1, 1/t is the conjugate of t.

Now, the n'th cyclotomic polynomial is simply the minimal polynomial (over Z) for t, hence also for its conjugate 1/t. Writing out the polynomial as

p(x) = ak xk + ... + a0,
we see that
0 = ak tk + ... + a0
0 = ak 1/tk + ... + a0.
Multiplying the last equation by tk, we get
0 = a0 tk + ... + ak.
So if q(x) is the polynomial with reverse coefficient sequence compared to p(x), we have that q(t)=0 and has the same degree as p(x), and is therefore a multiple of it. But a0 is the product of all primitive n'th roots of unity, hence is either +1 or -1. Thus either p(x)=q(x) or p(x)=-q(x), as required...