Unfortunately, Catalina's gatekeeper prevents a normal use of yEd at this time. You can try to bypass gatekeeper by running yEd from command line:
Install the latest Java runtime environment on your macOS (unless Java is already installed on your machine, of course). E.g. AdoptOpenJDK offers macOS installers for OpenJDK 13.
Download the "Zipped yEd Jar" yEd-188.8.131.52.zip from the yEd download page.
Extract the zip archive to a destination of your choice.
Open a terminal and start yEd with
java -jar [path-to-yed]/yed.jar
where "[path-to-yed]" is the path to the directory extracted from the zip archive in step 3 above (e.g. ~/Documents/yed-184.108.40.206).