Hi! Welcome!
The installation is getting stuck at the point when R tries to move the downloaded package files from a temporary directory into your R library.
The most common reason for this problem seems to be anti-virus/anti-malware software preventing files from being moved within your User directories (which is where your R library lives). This is usually meant as protection against ransomware.
As a test, you could try temporarily turning off any anti-malware software you are running, and then try installing again. If it works, then you’ll need to figure out how to change settings in the anti-malware software to remove these restrictions from RStudio and R. There are some pointers for a few popular anti-malware software packages in the comments on this StackOverflow answer: https://stackoverflow.com/questions/5700505/windows-7-update-packages-problem-unable-to-move-temporary-installation#5714458
P.S. It is unlikely this problem has to do with the RStudio IDE, so I’m moving the post to a different category to cut down on noise in the #rstudio-ide channel. You also have the power to recategorize your posts: FAQ: How Do I Change Categories, Tags, and Title?