Update: After extracting the files to any location (location does not matter, FileZilla can detect its own installation prefix), you can start the program using the filezilla executable in the bin/ subdirectory (wiki.filezilla)
Or compile FileZilla from source:
Download and uncompress the downloaded file.
tar -vxjf FileZilla_3.38.1_x86_64-linux-gnu.tar.bz2
Go to the newly extracted folder and install the missing libpng16 that FileZilla is requesting:
Ubuntu 17.10 and older
sudo apt install libpng16-dev
Ubuntu 18.04 and newer
sudo apt install libpng-dev
Setup the ubuntu-toolchain-r PPA for installing the correct version of libstdc++.so.6 that FileZilla is looking for.
sudo add-apt-repository ppa:ubuntu-toolchain-r/test
sudo apt update
sudo apt upgrade
Copy over the updated FileZilla files to their correct locations:
sudo cp -ravx * /usr/
Launch FileZilla as normal.