The reason you might want to shoot RAW photos on your phone instead of whatever your current phone offers really only comes down to the matter of control. Most phones output a file type that has removed much of the original information that is minimally visible in the image itself in order to reduce the size of the file for the sake of portability, after all the vast majority of people using a phone camera aren't doing so for precision images, but rather for preserving memories or sharing experiences.
RAW files contain much of that unseen data that allows you to bring details out from under-exposed, dark portions of the image, or to recover details from over-exposed, bright parts of the image.
There are a number of different ways you can shoot RAW, but almost all of them rely on a secondary app to capture with. Currently, the only in-phone option is on the iPhone 12 Pro and iPhone 12 Pro Max, and this is only the case if you have updated to at least iOS 14.3.
This is a fantastic option for both iPhone and Android users. It is available to anyone that has a Photography or higher level account with Adobe -- this is available for free for students, faculty and staff at Harvard.
Simply create your account with Adobe, and then download Lightroom App in your phone's mobile App store, and log in using your Adobe account. From here, you can make sure that the app is configured to shoot in RAW, using these instructions.
This is another great option for both Android and Apple users, just download it from your mobile app store and create an account. It will prompt you for a subscription, which you can skip, that simply offers a larger pool of filters for your photos.
The VSCO app is a good one for those without an Adobe account, but offers a far less robust control over editing your photo, although you can send the RAW image to another device to touch up in a separate app.
This one is an honorable mention, it is only offered for Apple users, and you can either pay a one-time $40 or $12/year.
It allows you to shoot in RAW, but if you're looking to do so regularly, it's real draw is in its simple and powerful UX design, as well as offering a set of tools that are closer to the experience of using a high-end DSLR.
Finally, for the extremely unlikely corner cases, SnapSeed is a Google-based free app for editing RAW images, although it unfortunately doesn't yet have functionality for taking RAW images. So if you don't have Adobe, but do have and iPhone 12 Pro/Pro Max, then this could be the perfect option for you. But will also work for anyone just looking for a great photo editing software, regardless of if your images are RAW.
It has a number of the same functions as Lightroom and the other apps above, it mostly just stands out as it's free and available on any operating system.