The Max Speed button was a feature available to everyone at the game's release, but the free trial was removed in the Android 1.5.2 and iOS 1.5.3 updates. Although it is still free on day 1 only. It can now only be used if you buy it for $2.49* or if you're a VIP user.
Alternatively, all users could have unlocked the Max Speed feature for free if they unlocked all character's After Endings and Secret Endings 1&2 by November 7th of 2016.
*Price in the USA. Price varies from country to country depending on the country's currency exchange rate with South Korea.
The Max Speed button can be used to quickly speed up the chat dialogue and effectively allows you to skip through chatroom dialogue much like how the 'Skip' button works for Visual Novel modes.