Liquidity trap and consumers' reaction to an increase in money supply I've understood that when the nominal interest rate reaches the zero lower bound there is nothing that the monetary policy can do to increase the output level of a certain economy. Because $i=0$, no one is interested in bonds anymore, and people decide to hold more money in correspondence to the same interest rate. But if they hold more money, why they decide to keep them, instead of increasing their level of consumption and thus increase also the level of production?

It may be a stupid question, but I really cannot figure out the answer. Hope you can help me. Thank you in advance.
Good question. The answer depends on what exactly you mean by an increase in the money supply and how it is implemented. 

Because standard monetary policy (ie. open market operations) is implemented through banks, it is functionally quite different from a pure infusion of cash to consumers, which is more the realm of fiscal policy, depending on how it is financed. Consider a central bank that buys bonds from commercial banks by crediting their accounts with reserves. Banks now have excess reserves which they are able to lend should they so choose. If interest rates are above zero, interest rates will fall enabling banks to offload the excess reserves. However, if interest rates are already at a nominal lower bound of zero, the price of loanable funds cannot fall to stimulate increased demand, and hence banks cannot offload the excess reserves. 

It also depends on whether the increase in the money supply is temporary or permanent. A permanent increase in the money supply will permanently increase the price level, thereby inducing a one-period spike in inflation and a contraction in real interest rates ($i^R=i^N-\pi$). The quantity of loanable funds demanded will increase. If the increase in the money supply is temporary, though, this will not work. This is why economists view the implications of permanent versus temporary changes to the money supply very differently during zero bound episodes.

A few links that may be helpful:

http://www.voxeu.org/article/combatting-eurozone-deflation-qe-people
http://www.bruegel.org/nc/blog/detail/article/1527-permanent-qe-and-helicopter-money/