It's not exactly full-screen, it's more like a maximized box that still lets you see your labels and your chat contacts. You can manually enable the "full-screen" mode every time you want to use it or set is as the default option.
"When this option is enabled, the compose window is centered in your inbox and expands to fit on your screen. In addition, the formatting toolbar is on by default. You can click on the expand button in the top right to switch to full-screen or set full-screen as the default by selecting Default to full-screen in the more options menu in the bottom right," informs Google.
There's a help center article that provides more information:
"Compose messages in a window that's right for you. The default is a small compose window that's great for checking other emails as you type and getting things done quickly. If you prefer a larger window, you can switch to full-screen when you need it, or set full-screen as default."
It turns out that the new expand button for the full-screen mode replaces the pop-out button, which opened the compose box in a new window. To open the box in a new window, you now need to shift-click the expand button.
The full-screen mode also works when you reply to a message, but it's more cumbersome to use. "To compose in a larger window, click the Recipient arrow next to the recipient's name and select Pop out reply. Once you've popped out the reply, click the expand icon expand to compose in full-screen."
As usual, this feature is gradually rolling out to all Gmail accounts, so check back later if you don't see it right away. Obviously, this only works in the new compose interface.