I also use eagle for many years. So, after you download the executable file eagle-5.6.0.run use chmod to mark the executable bit:
chmod u+x ./eagle-5.6.0.run
I always install eagle as superuser, so before running you may want to su for installing it in a directory such as /opt/eagle-5.6.0
To execute the file in your bash shell just type
./eagle-5.6.0.run
. This will bring up the installation dialog. Follow the instructions and eagle will be installed in your system.
If you try to execute (as the normal user of course!!!) the eagle binary in the installation bin directory (/opt/eagle-5.6.0/bin in my example) then it will complain that it cannot read the file eagle.key. To solve this you must (under su) move into the directory /opt/eagle-5.6.0/bin and
chmod a+w freeware.key
and then
cp freeware.key eagle.key
After that when running eagle for the first time just select the freeware.key file and everything will go fine. After the first run you may want to restore the access rights of the key files so as su issue the command