This is asking you to update the rlang package. Try resetting your R session (to make sure rlangis not already loaded) and updating rlang with this command:
install.packages("rlang")
Then you can try installing lifecycle again.
If you get any error message while installing rlang please post it here so we can take a look at what is going on.
Note: RStudio is not related to your situation since it is just an IDE for the R programming language and it has no direct effect on your R package library. That is why I changed your topic's category.