binomial()$linkinv(fixef()): what exactly does these function are for when applied to mixed generalized analysis?

I was workinf on a dataset, trying re-perfomrming an already run statysical analysis and I met the following function:

binomial()$linkinv(fixef(m))

after running the following model

summary((m = glmer(var1 ~ str2 + (str2 | ID), single_data, family = binomial)))

My first question is what exactly does this functions is made for? Beacuse throgh other command lines the reciprocal code as well as a slightly modified code based always on it are also reported:

1) 1- binomial()$linkinv(fixef())
2) single_data$fit = binomial()$linkinv(model.matrix(m) %*% fixef(m)) #also the sense of the operator %*% is quite misterious too.

Thanks for your answers

Solution

I agree with most of @Oliver's answer. I will add a few comments (since I had an answer partly composed already).

I would be very wary of the script you are following: some parts look wrong (I could obviously be mistaken since these bits are taken completely out of context ...)

binomial()$linkinv refers to the inverse link function for the model used. By default (which applies in this case since no optional link= argument has been specified), this is the inverse-logit or logistic function A nearly equivalent function is available via plogis(), but using $linkinv could be better in some cases since it would generalize to binomial analyses done with other link functions [e.g. probit or cloglog].
as @Oliver mentions, applying the inverse link function to the coefficients is at least weird, I would even say wrong. Researchers often exponentiate coefficients estimated on the logit/log-odds scale to obtain odds ratios, but applying the inverse link (usually logistic function) is rarely correct.
binomial()$linkinv(model.matrix(m) %*% fixef(m)) is indeed computing the predicted estimates on the link scale and converting them back to the data (= probability) scale. You can get the same results more reliably (handling missing values, etc.) by using predict(m, type = "response", re.form = ~0) (this extends @Oliver's answer to a case that also applies the inverse-link function for you).
I don't know what binomial_pred_ci is either, but I would suggest you look at predictInterval() from the merTools package ...

PS these answers all have not much to do with runjags, which uses an entirely different model structure. Presumably glmer models are being fitted for comparison ...