The concept of the liquidity trap goes back to Keynes and to Hicks' interpretation of the argument. For a long time, it was more part of a theoretical debate than a real possibility, but Japan's crisis in the 1990s changed this slightly.
To understand the "liquidity trap" is important to remember that it takes place in moments of highly recessionary environments. In these instances, nobody is really interested in investing much and the price level may be going down. The key insight is that monetary policy may end up being irrelevant in this situation. Why? Because there is a downward limit in the reduction of interest rates: when nominal interest rates are zero or almost zero, how can you reduced them more?
With interest rate equal to almost zero (with bonds at very high levels), the is no real reason to hold bonds. This mean that any expansion of money will result in an expansion of liquidity in the hands of people... and at the same time there are a lot of idle resources in the economy. You can find useful explanations of this process here and here.
Since the late 1990s, the concept entered into the debate on the Japanese crisis: in Japan, low interest rates were very low and monetary policy ineffective. Paul Krugman elaborated a dynamic model in which he showed that the liquidity trap can be a real possibility if people think that the price level will not increase steadily. You can find his explanation here (just read the introduction as the rest is somewhat complicated).
We represent the idea that monetary policy by a horizontal LM curve. Again, expansions of money supply do not affect the interest rate and will only result in more liquidity in the hands of people.
What about fiscal policy? In this environment, fiscal policy will be very effective both in the simple IS-LM model but also in the more complicated, Krugman version.
How can we see the intuition in our graph? Here is a possibility: there is a lot of liquidity (and a lot of idle resources) in the hands of households and firms. In this case, government spending will simply put some of this money "to work" but there will not be further increases in the demand for money (people are already holding a lot of money).
Is this an ideal world because you have fiscal policy with crowding out? Not really because we are in a deep recession and you may need to expand fiscal policy more than you are capable of (in terms of your borrowing constraint). Moreover, it could be that fiscal policy does not help to re-start the economy.
To follow all this discussion and the IS-LM model a little better, you can also check these slides from a professor at the University of Hawai.
Finally, how likely is the liquidity trap in practice? There is a lot of debate about it, but many people believe that we were very close to one in the recent global crisis, particularly in the US. Interest rates were very low, but this did not promote investment. People (and banks) simply kept more money in their pockets. At the same time, however, there has not been a process of deflation and the economy is slowly recuperated (although nobody knows for how long).
How likely are liquidity traps in developing countries (our main interest)? The answer is probably that not very likely because inflation is higher than in developed countries and interest rates can go down significantly. Moreover, as our Stiglitz et al book argues, the expansion of liquidity can have a positive effect on firms that are constraint by a lack of credit.