Why can ECC key sizes be smaller than RSA keys for similar security?
Actually, the problem is that the above quote uses the term "discrete log" in a way that's different from what you're thinking of.

When someone uses the term "discrete log", they can mean two things:


A discrete log in the group $Z^*_p$; that is, given $p$, $g$ and $g^x \bmod p$, recover $x$
A discrete log in some other group; that is, given a group $G$, a generator $g$ and the value $g^x$, recover $x$.


Of course, the precise meaning is given by context.

In this specific case, the quote refers to the first meaning of discrete log; a log over a multiplicative group modulo a prime (and for such a group, there exist subexponential ways of solving it).

However, when we're dealing with the security of an EC group, we don't care about discrete logs modulo $Z^*_p$; instead, we care about the discrete log within the EC group (that is, the second meaning).  These EC groups (or, at least, the ones based on curves that we actually use) do not have these fast methods of solving it; in particular, the number field sieve algorithm does not appear to apply.

And, because the DLOG problem in an EC group is much harder, we can use a smaller EC curve safely.