When you open a menu and close it, it allows you to gain movement for something, this could skip waiting for the egg animation at the beginning to be over so you can grab the apples, or you can skip days by being allowed to put the caterpillar to sleep again for every time you open the menu, so how do you do this?

On the newest version, just press the back button 2 times, 1 to get the menu to quit, and another to get out of that menu. After that, you can do whatever animation skipping related thing you want to do.