My question is what does this description includes?
The information that describes what elements are in the group, and how to perform the group operation. In a $\mathbb{Z}_n^*$ group, the value $n$ may be sufficient, for an Elliptic Curve group, that'd include the curve equation (e.g. for Weierstrass curves, the values $a$, and $b$ in the equation $y^2 = x^3 + ax + b$), and the finite field the group is defined over. Actually, in practice, we typically don't generate curves on the fly; it is far more typical to say "I'm using P256" or "I'm using Curve25519".
Is it because for example in RSA the group elements are known before
No, it's because that the group $\mathbb{Z}_n^*$ has additional properties that generic groups don't have, and an attacker can exploit these properties to make solving the discrete log problem faster.
One such property is that fact that we can construct a modest-sized list of elements $p_0, p_1, ..., p_i \in \mathbb{Z}_n^*$ such that, given a random element of $x \in \mathbb{Z}_n$, we have a nontrivial probability of being able to quickly find the values $e_0, e_1, ..., e_i$ such that $x = p_0^{e_0} \cdot p_1^{e_1} \cdot ... \cdot p_n^{e_i}$; we don't know how to do that with a generic group, and it turns out to be useful in the discrete log problem.
And, in case you're wondering what that quick algorithm is, well, we set $p_0, p_1, ..., p_i$ to the first $i+1$ primes, and given $x$, we treat it as an integer (rather than an element of $\mathbb{Z}_n^*$) and do a quick factorization on it, using an algorithm that is designed to be find small factors (and if it fails because of a large factor, we can detect that quickly). A nontrivial fraction of $x$ turn out to be smooth, and hence this works sufficiently often.
