How do the bits of a certain emoji code for all the colors?
That binary number does not contain the information about the pixels in the image which is used to show the emoji. It is only a number which, according to the Unicode standard, refers to that emoji. Every program which shows text (e.g. email reader, web browser, ...) must have a way to convert that number into the actual pixels. Usually, in modern computers, there is a font file, which contains all the information about the pixels of each character. So, when the program has to display the emoji, it accesses the file and reads the pixel data corresponding to the emoji, and sends those pixels to the screen.

Note that the Unicode standard does not specify what these pixels are, but only that they should represent, say, a "cake emoji". Every program might use a different font, showing the cake in a different way. 

This also happens for regular characters. Unicode says that "A" is represented by number 65, but there are many fonts that can be used to show that "A".