The imagery is either third party raster images, or specific renderings of the existing map data. It looks like way marked trails are the latter, so the data should already exist on the map.
Typically you would find those in JOSM by doing a search from the edit menu. E.g., in my part of London, searching for “type:relation type=route route=foot” will find the Capital Ring. The route= stanza may vary.
You can also use the waymarked trails image to identify one of the segments of the route and then right click on the relation in the containing relations list and do select relation.
If the relation doesn’t exist, but the individual roads and paths do, you will need to construct it. Unless you are very familiar with the copyright rules for OSM, you should do so by actually following the trail on the ground based on the sign posts. Do not do this with something like Google Street View, as that would be a breach of Google’s intellectual property rights. To be safe, you must see the signs in front of your own eyes. Trails in guidebooks are likely to be the copyright of the book publisher.
If some or all of the paths don’t exist, you will have to map them from first principles, e.g. from a GPS track or from acceptable imagery. Generally the imagery available in the JOSM imagery menu and the corresponding edit preferences, is acceptable, if you haven’t installed your own. The imagery may not allow you to identify the type of path, or road, without physically visiting it, or using some other imagery.
I’m not familiar with the tools for creating Garmin maps, so I don’t know how you go from selecting the relation to producing a Garmin map.