How do I perform mathematic operations on raster data in R?

You can do mathematics with rasters with the usual mathematical operators.

There are assorted functions in the raster package for doing grouped functions, like monthly averages over stack of rasters.

There's also the terra package which works with rasters and should be faster than raster.

Install the geodata package to download some WorldClim data

# install.packages("remotes")
remotes::install_github("rspatial/geodata", dependencies=FALSE)

library(terra)
library(geodata)
wc <- worldclim_country("Iceland",  "tavg", ".")

A fast and simple approach

bio1 <- mean( clamp(wc, 0, Inf) )

A more general, but slower, approach

biotemp <- function(x) {
    x[x < 0] <- 0
    rowMeans(x)
}

bio2 <- app(wc, biotemp)

A perhaps more direct approach is also possible

wc[wc < 0] <- 0
bio3 <- mean(wc)

This works fine small data sets that can be held in memory. But with larger datasets the performance deteriorates (perhaps even fails), especially with more involved computations as lots of temporary files may be created.