Hey unnona this is a good question and there is a few ways people get the foil look. If I was to guest, I think detritus did a very classic hot-stamping foil process for this, which involved a larger letterpress, usually air-feed, for production and requires larger runs and is a premium $$$ finishing technique.
The process involves making a metal plate, it is mounted on a base which is heated up and then there is a roll of coloured foil feed between the paper and metal plate, which is sandwiched between it all with pressure to transfer. There is lots of foil waste, the plates and set up are expensive and it requires a shop set up for it. A die-cutting company, or a larger press in your area would likely offer this service.
Example [https://www.youtube.com/watch?v=-dNQncKhTu0]
Alternates are you can find smaller machines with smaller print areas that heat up lead type you hand set and you apply pressure like a drill press to the foil over the paper to Deboss an leave foil behind. Any old hard over books or book restores will likely have this process done to identity them.
Example: [https://www.youtube.com/watch?v=g1qrRht6MtU]
Both of these methods give consistent, beautiful results, with a slight debossing to the paper but the first would requiring big runs or big per unit costs. The second is all hand work and requires machines that are hip and therefore in demand even used.
An alternative you can find different Chinese import machines hat do a foil heat-transfer technique. You print your image on a toner printer, then run it through hot rollers (like a laminator) with foil rolls that have glue on one side. The heat activates the glue and toner and when it sets the foil is peeled off leaving foil where your toner has been. Does not work with oil or water based inks, just toner because toner is heat set. Results are inconsistent across batches but it is the cheapest way to get a similar result.
Example: [https://www.youtube.com/watch?v=xLMpe3sZAr4]
On that note, I have a heat transfer foil machine and love working with it. If perfection isn't required it is affordable and has a small footprint. Recently a friend has been experimenting with doing the same process by running foil and toner printer paper through a laminator to apply heat or just ironing it with some parchment paper or a towel between them. Much more hand work but for small ziner runs or the cheapest option this is a possibility. The quality across the run will be pretty inconsistent.
One other option is on large trade printer web presses some have invested in a product called aquafoil. Basically you print in a 4-colour press, then put a 5th "colour" down which is a transparent thing called aquafoil which makes the underlying image glossy and foil-like. Again a premium product but if you are sending out a job could be competitive with a custom hot-stamp process.
Hope these descriptions and videos help you visualize it.