Linux "installation" is just adding the location of a file to the PATH environment variable after you ready the executable for a program. If the shell can find an executable file with the same name as your command on a directory listed in the PATH variable, it will execute that file.

Here, you're adding the path to the trinity binaries to an environment variable named TRINITY_HOME. Let's say you downloaded the tar.gz file to ~/Downloads and then built it, the binary would be in some location under your current working directory (~/Downloads/Trinity=*). I'd recommend running find $PWD -name "Trinity" -type f. The manual says you should see the file in the directory you extracted the tar.gz contents to, so the find may not even be required.