The message and the documentation are accurate. The message name itself is slightly inaccurate, since it's really full page instead of fullscreen.
"Show page" means that the button transitions from ApiSandbox covering the whole browser window to showing MediaWiki's normal page chrome.
I am utterly confused. Message and description still appear contradicting to me, and I do not understand the " browser window ... page chrome " - sorry. So what am I supposed to translate the message to?
When you load https://en.wikipedia.org/wiki/Special:ApiSandbox, it displays widgets inside the normal page layout. There's the toolbar and tabs across the top, the sidebar on the left, a title, and so on. This is the "page chrome".
I have no idea what you should translate that to, since the only non-English language I know is Spanish and I don't know enough of that to do translations. You should probably localize it by looking at what the buttons do, and then determine what an appropriate button text would be to indicate that.