1) "To first order" appears to be a shorthand for "to a first-order approximation", meaning in turn, "to a linear approximation".
2) It is not that "the one implies the other", it is that the one is consistent with the other.
In attempting to arrive at a functional specification for the "tax distortion costs" function, say $C_D=h(T)$, first we reason that if tax revenue is zero (not due to inefficiency of tax collection and/or tax evasion, but because the tax rate is set to zero), then distortions due to tax will also be zero. This implies that $h(T)$ should not contain a constant term, so that $h(0) = 0$.
Moreover if we reason (as D. Romer does) that "distortions by taxes are likely to increase more than proportionally with the amount of revenue raised", this implies that $h(T)$ should not be linear in $T$, but it should be convex (this is what it means to say "more than proportionally").
The natural step then is to consider a quadratic distortion cost function, without a constant term:
$$C_D = aT^2,\;\; a>0$$
which reflects both our theoretical arguments (and in particular, $h(0) =0$).
But this also implies that "distortion costs are zero to a first-order approximation", because we have
$$h'(T) = 2aT \implies h'(0)=0$$
so a first-order approximation (Taylor expansion) around zero would give
$$h(T) \approx h(0) + h'(0)\cdot T = 0+0\cdot T =0$$