We have $$C_{n}^{p+t}C_{n+p+t}^n=[y^{n-t}x^tz^t](x+y+z)^{n+t}\cdot \left(\frac{(x+y+z)y}{xz}\right)^p,$$
where $[M]f$ is a coefficient of monomial $M$ in the polynomial or series $f$. Multiplying by $(-1)^p/p$ and summing by all $p=1,2,\ldots$ we get
$$\sum_{p=1}^\infty (-1)^p p^{-1}C_{n}^{p+t}C_{n+p+t}^n=[y^{n-t}x^tz^t] (x+y+z)^{n+t}\log\left(1+\frac{(x+y+z)y}{xz}\right).
$$
It remains to use the identity 
$$
\log\left(1+\frac{(x+y+z)y}{xz}\right)=\log\left(1+\frac{y}x\right)+\log\left(1+\frac{y}z\right)
$$
to get twice the right hand side of your formula.

We should explain what is the ring the series above belong to, let it be the ring of power series in $x,z$ and $y/(xz)$.