面 by itself means "face", while 白 by itself means "white". How did these two words combine together to mean "interesting"?
According to 語源由来辞典 ( http://gogen-allguide.com/o/omoshiroi.html ), 「面白い」 is originated from 「面白し」. 「面」 used to mean "a sight/view" (the source says the front of eyes) and 「白い」 used to mean "bright and clear." Then 「面白し」 later came to mean "a light/bright sight/view" and then later "a beautiful sight/view". It further came to mean "fun" or "comfortable", which represents a pleasant feeling.
For your information, "interesting" better translates to 「興味深い」 to be accurate although it is often roughly translated as 「面白い」. 「面白い」 has several meanings with different nuances.
The accepted origin of 面白い (omoshiroi - interesting) goes as follows.
面 (omote - the front) means whats in front of you.
白い (shiroi - white) means bright and clear.
These combine to mean that the scene in front of you is bright and clear. This first took on the meaning of beautiful scenery and then was later used to express a scene that is fun and enjoyable.
Thus, over time 面白い came to mean interesting.
There is another origin story which is probably based more on nostalgia for the past rather than facts. It goes as follows.
In past times, people often spoke around the fire at nighttime. When somebody said something interesting, the others' smiling faces would rise up and appear white in the light of the fire. Thus, interesting talk leads to the whiteness that appears before one's eyes.