This extension's language is called "Celeste Dialog Document", but targets .txt files. You may need to manually switch to this language whenever you create a file, or if Visual Studio Code doesn't detect it automatically. There are multiple ways to do this:
When you create a new file, Visual Studio Code should let you know that you can manually set the desired language.
The currently selected language is displayed at the bottom right of the window. For .txt files, it's usually something like "Plain Text", but it varies depending on the extension of the file. Clicking on it will allow you to change the file's language regardless of its extension.
Use the command palette (brought with Ctrl+Shift+P) and search for the "Change Language Mode" setting. This brings a dropdown menu from which you can look for the Celeste Dialog Document language.
You can configure Visual Studio to always associate a .txt file with the Celeste Dialog Document language within the scope of your project, or globally.
To do so, open up the Preferences Settings tab (through File > Preferences > Settings, or with Ctrl+,).
Look for the "file association" option. You can choose whether you edit this setting globally or for this workspace only, by clicking on the corresponding tab below the search bar.
Then, add an entry to associate .txt files with the Celeste Dialog Document language. The key should be *.txt, and the value should be celeste-dialog. If your workspace is a the root directory of a mod, you can specifically target Dialog files only, by setting the key as Dialog/*.txt.