There are several ways to approach this problem.
First, be aware that not all players are curious by nature. Some people simply do not feel the drive to ask questions about their environment or to pursue interesting threads. Other people are unceasingly curious and will question everything.
My current group has two of the former and two of the latter. I am finding that I need to tone down my descriptions of locations, because even a casual mention of pipes on the wall, intended simply as flavor while the group makes its way through an abandoned waterworks, prompts a ten-minute detour as the curious players examine every aspect of the pipes. Now, my uncurious players are often prompted to act based on the investigation of the curious ones, so it may be that your group lacks an instigator whose curiosity will bring the others out of their shell.
If this is the case and you have an entire group full of non-instigators, you need to either find or create an instigator. You could add a new player whom you know to be curious (granted, not a very simple or easy solution, or even an applicable one in many cases). Or you can create an instigator, by finding out what interests one of your players and creating a plot hook for that player to latch onto. For example, in my previous game, one of the characters had a tragic backstory involving a dead little sister. I had a mysterious note delivered to him from the supposedly dead sister, and he was off like a shot, dragging the rest of the party with him.
Second, make sure you are giving your players a story they will be curious about. If your villain is attempting to steal a relic or destroy a village, your players likely have little reason to care. However, if the villain is attempting to steal the paladin's ancient holy relic to use in a foul ritual of undeath, or is leading an army whose next target is the bard's home village, the situation is now personal. The players have reason to be invested in the plot, and they will begin asking questions.
For example, also in my previous game I had a fighter who was just quiet - stayed in the back of the party, kept to himself, never spoke up. I'd originally hooked the party by offering a large reward for the retrieval of the relic stolen by the villain, but this character just didn't care. He went along with it because he was basically a good guy and his friends were doing it, but he had no personal interest and so didn't participate much. So I had the living personification of the relic appear to him (and only him) in visions. As he got to know her, not only did the loss of the relic become personal to him, but he also started asking questions at the prompting of the other players, because he was the only one the personification would appear to.
Third, be aware of how you comport yourself as GM. If you do find yourself monologuing, then stop. Give your players a basic description of the area/situation, then put on an evil GM grin, sit back, and say nothing more.
Silence is sometimes the hardest part of running a game. You have an awesome story you want to tell, so you try to tell it - but in the telling, you don't leave enough room for the players to find their footing. To stop talking is almost certainly going to create some awkward silences at first, especially if your players are accustomed to you eventually caving and talking more. So if they don't speak up, prompt them; say "What do you do?" and then, again, stop. Wait.
There's often a bit of acting involved in GMing, and I'm not just talking about portraying PCs. You must give the impression that you are a) confident, b) in control, and c) planning something. Silence is a powerful tool for portraying all these things; silence says "I have given you everything, and I expect you to act now". Silence puts the ball in the players' court; they will learn quickly that things will not progress if they don't speak up.
A final consideration: If you try all the suggestions in all the answers and you still find that your players simply won't ask questions, it's probably time to talk to your players about why they aren't speaking up or interacting. Then work with them, both in and out of game, to resolve their concerns, whatever they may be.