Interpretation of a $P$ value of a categorical variable? glm(formula = seagulls$attack ~ seagulls$bird_sex, family = "binomial", 
    data = seagulls)
Deviance Residuals: 
   Min      1Q  Median      3Q     Max  
-1.310  -1.218   1.050   1.138   1.138  
Coefficients:
                   Estimate Std. Error z value Pr(>|z|)
(Intercept)         0.09403    0.16403   0.573    0.566
seagulls$bird_sexM  0.21300    0.23243   0.916    0.359
(Dispersion parameter for binomial family taken to be 1)
    Null deviance: 412.88  on 299  degrees of freedom
Residual deviance: 412.04  on 298  degrees of freedom
AIC: 416.04

How do I interpret the $P$ value of the intercept? Does it mean seagulls$bird_sexF has this $P$ value and we reject the hypothesis of female seagulls having a higher probability of attack?

I'm guessing you're trying to estimate the probability that a seagull attacks...something, conditioned on its sex.  The model is
$$ \operatorname{logit}(p) = \beta_0 + \beta_1x $$
Here, $x$ is a binary indicator for sex (1 if male, else 0).  The null hypothesis for $\beta_1$ is
$$ H0: \beta_1 = 0$$
versus the alternative
$$ HA: \beta_1 \neq 0$$
Failing to reject this null means that, at least from these data, we can not conclude that seagull sex has an effect on the propensity to attack.  It may very well be the case however that with ore data, the effect would become significant due to the standard errors shrinking like $1/\sqrt{n}$.
