That value is obtained by taking the +1/2 hbar . omega term in each mode and consider all possible modes up to some cutoff like the Planck length. As such, you get an incredibly high energy density.
Of course, as general relativity allows for a cosmological constant which has about the same function, one could say that this cosmological constant sets off exactly in the opposite direction so that the net effect is what we happen to find as a background density. But that begs the question of how it could happen that there is such a "fine tuning" of both.
The usual trick that is done in this case is to postulate "normal ordering" of the operators, which amounts to putting the +1/2 hbar.omega term in the dustbin. However, in how much this is something deep, and in how much this is a cheap mathematical trick to get rid of it, can be debated.
Probably you have to think of it as this way: QFT is not the right theory to try to estimate the energy density of the vacuum (which only plays a role when general relativity is in the game, which isn't included in QFT in the first place). In other words, calculations of the energy density of the vacuum fall outside of the scope of validity of QFT as we usually know it. A bit like saying that calculating emission spectra of atoms falls outside of the scope of Newtonian mechanics.
You might be shocked to learn that that most sophisticated theory, QFT, has things "outside of its scope". Well, to paraphrase Witten (or was it Schwarz ?) half jokingly: with a $1,- experiment, you already show something that is not explained by QFT: take a cheap pen. Drop it. See it fall. There you are. (in fact, this is from memory, and he said something like "with a $1,- experiment, you can show something that string theory predicts...)
