By default, Poedit does not have the ability to parse Twig templates. In Twig documenation there is a simple example which shows how to extract translations. This can work with few modifications, but I wanted to make reusable tool which will allow you to parse Twig templates just like you do with plain PHP templates.
The result is the Twig-Gettext-Extractor that's set up inside Poedit as a parser command for "*.twig" files. This automatically extracts the information so that the editor can read the data from the template.