I think I may have reached an understanding about this. The summary of the fixed effect coefficients from the model is (on the scale of the link function, here log-odds):
summary(fit)$coefficients
Estimate Std. Error z value Pr(>|z|)
(Intercept) 0.206 0.222 0.927 0.354
sexM -0.412 0.289 -1.426 0.154
These are the same estimates as when averaging over the random effects for intercept (the sex effect is constant). So, these fixed effect estimates (on the link scale) are population average effects:
coef(fit)
$classroom
(Intercept) sexM
class 1 0.1605 -0.412
class 10 0.3606 -0.412
class 2 0.2119 -0.412
class 3 -0.0294 -0.412
class 4 0.0281 -0.412
class 5 0.0855 -0.412
class 6 0.2237 -0.412
class 7 0.3090 -0.412
class 8 0.2119 -0.412
class 9 0.4990 -0.412
colMeans(coef(fit)$classroom)
(Intercept) sexM
0.206 -0.412
However, when the coefficients are transformed to odds-ratios this is no longer the case:
sapply(coef(fit)$classroom, exp)
(Intercept) sexM
1.174 0.662
1.434 0.662
1.236 0.662
0.971 0.662
1.028 0.662
1.089 0.662
1.251 0.662
1.362 0.662
1.236 0.662
1.647 0.662
colMeans(sapply(coef(fit)$classroom, exp))
(Intercept) sexM
1.243 0.662
The intercept above (averaged over the exponentiated random effects) is not the same as the exponentiated fixed effect coefficient below, because exponentiation is not a linear transformation. But, the sex effect is the same, since this did not have a random effect associated with it. Therefore, the sex coefficient is a population average estimate, even as an odds-ratio:
exp(fixef(fit))
(Intercept) sexM
1.229 0.662
Best Answer
The fixed coefficients in GLMM are subject specific estimates, which mean the association between $X$ (the predictors) and $Y$ (the outcome, refer to the latent one in nonlinear models) holding constant the random effects, and it describes how an individual's mean of outcome depends on the predictors. You can find the clarification of subject specific and population average estimates here.
The importance of fixed coefficients depends on your goal of analysis. The fixed effects are often the focus, unless you are interested in the individual trajectories (random effects) or the scale (variance) of the random effects.
The significance here (mostly based on $z$-test) has the same meaning as that in other tests, which means that you can reject the null hypothesis that fixed coefficient $\beta=0$.