Unfortunately because every setup is different there can be several reasons why OpenRefine doesn't start successfully and there is no single answer or solution that will fix it.
To do analysis on a problem like this and offer potential solutions we really need more information such as:
Operating system
Version of OpenRefine (and if Windows did you use "with embedded Java" download option or not)
Any messages in the OpenRefine console (if you are running on a Mac you need to start OpenRefine in a different way to normal to get the console information - this is explained under the "Mac" instructions at
https://docs.openrefine.org/manual/running#starting-and-exiting - see the bullet pointed list headed "To run OpenRefine using Terminal:"
This last is usually the place to start with troubleshooting as it will expose any obvious problems at least.