There are quite a few ways to get the images stored on a Game Boy Camera on your PC or Mac. Some methods are commercial ready-to-use, while others are do-it-yourself and may require soldering and/or knowledge of Arduino IDE. Most of the solutions behave as a Game Boy Printer emulator and require you to print the photos while some others require you to dump the cartridge RAM.
Commercial (ready-to-use) Solutions
The BitBoy was released in 2015. While it is not the cheapest solution, it is very easy-to-use. It connects to a Game Boy using a link cable and acts as a Game Boy Printer emulator. Images are stored onto an SD card. It can save many Game Boy Camera images in sequence using the batch print function. BitBoy can also save images from other games with Game Boy Printer support. It has an internal battery and charges over micro USB so it's easy to use on the go.
Game Boy Camera WiFi Printer
The WiFi Printer was released in 2019. Similar to the BitBoy it operates as a Game Boy Printer emulator and supports batch printing. It serves a minimal web interface which can be accessed by connecting to its WiFi hotspot. The web interface allows applying some color palette changes and downloading the images to your local device. This WiFi Printer is not battery powered and requires external power over micro USB to run.
Cartridge readers are generic devices that allow dump the contents of a cartridge RAM (.sav) or ROM (.gb). In the case of the Game Boy Camera, the RAM dump (.sav) would contain the photo data. One benefit of using a cartridge reader is you may be able to retrieve photos that have been marked "deleted" via the Game Boy Camera while the data has not yet been overwritten. The dumped .sav file can be processed to retrieve photos with a variety of tools including, the Game Boy Printer Web App (see below). Some cartridge readers have a dedicated function to dump Game Boy Camera photos. Most cartridge reader software runs on a PC so these devices are not a portable solution. These devices can also be used to backup other save games from other Game Boy games, so it can be pretty useful.
Submodule GB-01 - Dedicated Game Boy Camera exporting. ROM and RAM dumping, no writing function.
Mofosynes Printer Emulator
This project has been around a long time, and has seen a lot of great improvements. It uses a simple arduino nano with a Link cable attached. To download the photos to your PC you connect the arduino to your PC and the Link cable to the Game Boy and use the print function on the Game Boy. You then catch the serial output of the arduino (with the Arduino IDE for example) and paste it in Mofosynes Webtool to decode the photos.
While this method is very cheap (an Arduino Nano goes for less than $10), it is not very portable and requires quite a few steps.
Herr Zatackes GB Printer Web app
Using an ESP8266 wifi enabled board, Herr Zatacke has created a tool that can be used on the go and has it's own web interface. It has the option to use a small display as well. The web interface also is available on GitHub.io and can be used with the Mofosyne Printer Emulator or the .sav files dumped with a cartride reader.
The ESP8266 by default needs USB power, but if you are creative you can use a USB charging board with a battery and design your own enclosure for it all like on the photo below.
This tool is the most advanced and versatile option available. By far. It even offers synchronization with dropbox or a private GitHub repo.
Using original hardware
For the purists among us, it is possible to download the photos to your PC with original hardware. The result isn't great, and it's a lot of work though.
For this method you can put the Game Boy Camera cartridge in a Super Game Boy and use a Super Nintendo and a video capture card to make snapshots.