= Solution
The <negative binomial regression> assumes <independent random variables> conditional on their recorded days, with
$$
Y_i\sim\operatorname{NB}(r,\lambda_i),\qquad \log\lambda_i=\beta_0+\beta_1d_i.
$$
The <logarithmic link function> applies to the conditional <expectation>, not to the observed count. The displayed <maximum-likelihood estimates> are
$$
\widehat\beta_0=2.778,\quad \widehat\beta_1=-0.003169,\quad \widehat r\simeq195525.9,
$$
and the <dispersion parameter> for the fitted family is one. Thus the fitted <variance> is $\widehat\lambda_i+\widehat\lambda_i^2/\widehat r$, practically the <Poisson distribution> <variance> at these means.
Increasing the day by one multiplies the conditional <expectation> by $e^{\widehat\beta_1}\simeq0.996836$, a decrease of about $0.3164\%$. Comparing days 365 and 1 gives $e^{364\widehat\beta_1}\simeq0.3155$. The negative slope has <Wald statistic> approximately $-35.79$, so under the stated model assumptions there is very strong evidence of a decreasing conditional mean:
$$
\boxed{\widehat\lambda(d)=e^{2.778-0.003169d}.}
$$
This is an association with calendar day, not a <causal effect> established by the regression. The printed negative-binomial residual deviance is $7544.3$ on $1298$ <residual degrees of freedom>, which is a reason to investigate the adequacy of the assumed count model and dependence structure. In particular, the small reported slope <standard error> cannot make a misspecified model reliable.
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