docs: add documentation for the Next.js app router

[missionmike]

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Changes

While setting up Flagsmith for the first time in a Next.js v15 project, I had trouble getting it going because I couldn't find docs specific to the App Router. I could only find docs for the Pages Router.

After I worked through setting it up, I thought I'd contribute back to the docs to fill in the gaps.

How did you test this code?

This code is tested manually.

Testing the Docs

1. cd into docs folder, then run npm i followed by npm run start:

2. Visit: http://localhost:3000/clients/next-ssr
3. View changes, specifically:

and:

Testing the Code Help Section

To test the code help section, run these steps:

1. cd into frontend folder, then run npm i followed by npm run dev:

2. Then visit the app at http://localhost:8080 and sign in (I used Google auth).
3. Create an empty new project, then visit the "Features" tab.
4. Expand the "Code example 2" tab and select "Next.js (app router)" to view the new app router version:

5. Confirm that the original Next.js version is still there, relabeled as "Next.js (pages router)"

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[matthewelwell]

Thanks for the contribution @missionmike! I've added a few surface level comments and will defer to @kyle-ssg (who may not have time to look until tomorrow) for the technical side of things.

[kyle-ssg]

Excellent contribution, thank you so much! @missionmike