Why does not being able to draw a card make you lose (in universe)
Put simply, losing due to trying to draw from an empty library represents succumbing to madness.

In the flavor of Magic, the library basically represents memories and knowledge that you draw on to cast spells. In turn, mill effects are often represented as causing insanity or memory loss. For example, the first printing of the card for which "mill" is named, Millstone has this flavortext:


  More than one mage has been driven insane by the sound of the millstone relentlessly grinding away.


You can see similar flavor in the names of cards like Mind Grind and Mind Funeral.

Keep in mind that in general, mechanics are only loosely related to flavor, if at all. Not all mill cards are directly related to this flavor, but many have some relation to madness, or mental pain, or some other (generally unpleasant) interaction with the mind.