It might help to identify $Sym^r(\mathbb P^1)$ with the Hilbert scheme of degree $r$ effective divisors on $\mathbb P^1$. The Hilbert scheme represents a functor, as does the space $\mathcal M_g(1,d)$, and so you can (try to) construct a map from one to the other by thinking in terms of the functor it representsfunctors they represent.
As to why $Sym^r(C)$ is the Hilbert scheme of degree $r$ effective divisors when $C$ is a smooth projective curve, this is a (non-trivial, it has always seemed to me) exercise in making contact with reality from the somewhat more abstract world of flat families.
It might help to identify $Sym^r(\mathbb P^1)$ with the Hilbert scheme of degree $r$ effective divisors on $\mathbb P^1$. The Hilbert scheme represents a functor, as does the space $\mathcal M_g(1,d)$, and so you can (try to) construct a map from one to the other by thinking in terms of the functor it represents.
As to why $Sym^r(C)$ is the Hilbert scheme of degree $r$ effective divisors when $C$ is a smooth projective curve, this is a (non-trivial, it has always seemed to me) exercise in making contact with reality from the somewhat more abstract world of flat families.