Add NYT Games documentation (#34868)

* Add NYT Games documentation

* Tiny tweaks

---------

Co-authored-by: c0ffeeca7 <38767475+c0ffeeca7@users.noreply.github.com>

source/_integrations/nyt_games.markdown

@@ -0,0 +1,29 @@
+---
+title: NYT Games
+description: Instructions on how to setup NYT Games in Home Assistant.
+ha_category:
+  - Sensor
+ha_config_flow: true
+ha_release: "2024.10"
+ha_iot_class: Cloud Polling
+ha_codeowners:
+  - '@joostlek'
+ha_domain: nyt_games
+ha_platforms:
+  - sensor
+---
+
+The [NYT Games](https://www.nytimes.com/crosswords) integration fetches data about your progress on their daily puzzles.
+
+## Prerequisites
+
+Before setting up the integration, you need to fetch the token from the dev tools of your browser.
+
+1. On your computer, go to [NYT Games](https://www.nytimes.com/crosswords).
+2. Login with your account.
+3. Open the developer tools via right-click or by pressing F12.
+4. Open the network tab and refresh the page.
+5. Select a request with `.json` in the name and go to the cookie tab.
+6. The token can be found in the `NYT-S` cookie.
+
+{% include integrations/config_flow.md %}