If checked, scripts may select a custom version of the library by appending literally @${BRANCH_NAME} in the @Library annotation, to use same SCM branch name of the library codebase as that of the pipeline being built.
If such branch name is not resolvable as an environment variable or not present in library storage (SCM, release artifacts...), it would fall back to default version you select here.
Keep in mind that while the markup for such variable version specification is intentionally similar to what you would use in pipeline Groovy code, for simpler use and maintenance, the actual strings are expanded by the plugin as it pre-processes the pipeline script before compilation. Tokens spelled in the @Library annotation are not Groovy variables! The values substituted here are not influenced by run-time interpretation of the pipeline script source text!