This is a frequent issue faced by most of the developers trying to install GEOS, Cartopy and Proj.
It happens to be that there are certain packages that need to exist before installing GEOS or Cartopy. A quick look at your screenshot of the error, you can try out the following...
sudo apt-get install libproj-dev proj-data proj-bin
sudo apt-get install libgeos-dev
sudo pip install cython
sudo pip install cartopy
sudo apt install python3-dev
An alternative solution in case above does not work for you, the reason could be of a mixed channel problem in Anaconda, to resolve that, issue following commands as root (admin)
conda config --set channel_priority strict
edit the ~/.condarc file which will look like this after the edit:
channel_priority: strict
channels:
- conda-forge
- defaults
Use the root user to install it because it will have to access places where your standard user does not have access.
conda install geoplot -c conda-forge