This is going to sound really snotty.
Tell good stories.
If the story is gripping and tension-filled, the players will react.
See? It's easy! Heh.
Of course, telling good stories is the hard bit. Good stories need, first and foremost, good characters. I'm not talking about the PCs, here -- I'm talking about your characters. Do they live? Are they real, with personalities and fears and hopes? If they are, is that communicated to the players?
Your responsibility is to make the players care. That's what makes a great DM. You have to make the world come alive -- they won't do it for you. You have play the people in the world as REAL people. You have to act, you have to be smart and creative and inspirational and ruthless and funny and serious and you have to SHOW them how much it matters. It's hard. But you can LEARN how to do it.
The very fact that you WANT to do this tells me you've got what it takes. All you have to do is care enough to do the work that's required. Here's a few tips:
PLAY your NPCs, don't DESCRIBE them. That is, when the PCs walk into the bar, don't say, "The tavern doesn't allow swords, so you have to leave them at the door." Say, "You walk in, and there's a big bald guy at the door. He glares at you. 'Leave your swords here, kids.'" Now the big bald guy is a character. He's tough and used to being obeyed and isn't impressed by our heroes. Whatever the players do at this point, they've already got an idea how big baldie will react. They already have a relationship with him.
Have things happen while the PCs aren't around. They go away and come back -- things should be different. The political situation tranforms due to a peasant rebellion, or a priest is assassinated, or the Duke's daughter runs off with an elven ranger. None of these things require the party's attention, but the fact that they're going on gives your world more depth. And if they know that the Duke's daughter ran off with an elven ranger, they'll feel like the war that erupts between the Duchy and the elven kingdom next door is something that is happening not just because you want them to get into a war, but because it's a natural progression of what's been going on. It's real, not contrived.
Make sure there are consequences to everything they do. Everybody's got families, people who care about them. You kill someone, you're going to create enemies. You save someone, you're going to get friends. The issue here is creating a world where the players realize it matters what they do. They make a difference, for good or evil.
There are much worse things that can happen to a person than a 15d6 cone of cold. Driving someone insane, possessing them, turning them into a lycanthrope or a host of other possibilities. Making an enemy of King Whosis can make your life a short living hell. Players can find out pretty fast that they actually do care what happens when something like that hits them.
Beyond all this, create a detailed world with lots of cool stuff in it, create brilliant, believable characters that your players love to love or hate, come up with fiendishly twisted plots that turn their brains inside out, scare them, make them laugh, make them furious and give them memories they'll treasure.
Do that and you're halfway there already.