What does this mean?
Consider this scenario: You have a Living End in your graveyard, and you cast a Snapcaster Mage, and use it to give the Living End flashback. Can you then proceed to cast Living End from your graveyard?
The answer is no, you can't. And the reason for this is the principle: A non-existent mana cost cannot be paid.
Living End has no mana cost. This is not the same thing as having a mana cost of 0. (If its mana cost were 0, then it would have a mana cost symbol with 0 on its upper right corner.) It has no mana cost at all. This means that it can't be cast in the normal way (because paying its mana cost is part of the casting process, and in this case that step cannot be done, because it has no mana cost.)
(Note that Living End has a converted mana cost of 0. However, this is a completely different and unrelated thing. That value is used only for effects that look for converted mana cost explicitly. Casting does not look at converted mana costs.)
Snapcaster Mage adds flashback to the card when it's in your graveyard, and it states that the flashback cost is its mana cost. However, as stated, the card has no mana cost! This means that it has no flashback cost either, and a non-existent cost can't be paid. Therefore you can't cast it using Snapcaster Mage either.
So how exactly is Living End cast, then? Using its suspend ability. Suspend works like this: If the card is in your hand, any time you could cast the card (ie. obeying timing restrictions for the card type) you may pay the suspend cost, which moves the card into exile with the specified number of time counters. At the beginning of your upkeep you remove one time counter from it, and when the last time counter is removed, you may cast the card without paying its mana cost.
That last part is the crucial difference: The effect allows you to cast the card "without paying its mana cost". Any effect that uses that wording bypasses completely the mana cost of the card, and thus we don't even look at what it is (or, like in this case, whether it even exists.) The effect instructing you to cast the card doesn't require paying any mana cost, and thus the card can be cast.
Consider Living End with these two cards:
Act on Impulse exiles the top three cards of your library, and allows you to cast them (from exile) until end of turn. Even though the cards are in exile rather than in your hand, the casting process is identical. Therefore you can't cast Living End this way, because you can't pay its mana cost. (Incidentally, you can't suspend it either, because the suspend ability can only be used if the card is in your hand.)
Chandra's third ability exiles the top ten cards of your library. You then choose an instant or sorcery card among them, copy it three times, and then you may cast these copies without paying their mana costs. This, thus, does allow you to cast (the copies of) Living End, as mana cost is irrelevant.
As an unrelated side note, notice that these two effects behave different with regards to timing restrictions.
With Act on Impulse you have to obey the timing restrictions of the card (in other words, if the card is a sorcery or a creature without flash, you can only cast it whenever you could normally cast a sorcery, ie. in your main phase when the stack is empty.) With Chandra's third ability, however, you cast even sorceries immediately, ignoring their timing restriction (ie. even if the stack is not empty, and even if this somehow wasn't your main phase.)
The reason for the difference is that in the former case the effect allows you to cast the cards during a time period (ie. "you may cast these cards until (some time in the future)"), while in the latter case the effect instructs you to cast the cards immediately, while the effect is resolving. Whenever an effect instructs you to cast a card immediately, you do so disregarding the card's type; whenever an effect allows you to cast a card during a period of time (ie. until end of turn or whatever), you have to obey the timing restrictions of the card type.
