Interpreting the coefficient of a log-transformed variables is reasonably straightforward: it represents the predicted change in the dependent variable for a 1-log-unit change in the independent variable.
Here, the dependent variable (in the default log-link for the quasipoisson family in glm) is $log(y)$. So for the relation of $x_1$ to $y$, with the other independent variables held constant, you have a change of 1 log unit in $(1+x_1)$ corresponding to a change of 0.76 in $log(y)$. That pesky 1 in $log(1+x_1)$ makes is hard to provide a more general direct relation between $x_1$ itself and $y$.