Human beings are natural story tellers. We love them and more to the point, we need them. Think about it. How many stories do you tell in your average day? When you get home from work, you probably tell someone a story of your day. When talking to friends you'll tell the story of what you did recently. If you're a gamer - and if you're not, how did you get here? - you likely tell a lot of stories about what has happened in games past. Stories are important in your games as well. Today, I want to talk about why.
The first thing a story will give your game is motivation. This can be as complex as repaying political debts and clearing a familial obligation to as simple as "get the loot, become famous, win at life." This is the obvious benefit for a story, and yet a lot of times people forget about it. Especially in one shot games or other quick things. You ever have a player ask "why are we doing this?" That person has lost track of the story, the motivation, and thus isn't sure what is going on anymore. Yes, they can figure out that the party is stomping monsters and so they should help, but without knowing the 'why' their fun is already impacted negatively.
Along with motivation, narratives give us an explanation for what is going on. This works towards providing the motivation, but it also offers up other details. Think of most of the games you've played, I'll bet that any one where things weren't explained properly stick out a bit more than others, right? Why? Because they weren't explained so your brain may still be noodling them.
Even old school games that are famous for their brainless action (Doom, Wolfenstein 3d, etc) had a story to give explanation. Sure it was a simple thing stapled on, but it gave an explanation and meant you weren't there going "wait, why are all these infernal imps here?"
The final big thing that a story can give is flow. It helps to pace things. It gives you breaks and moments of tension. It also just lets everything easily flow out and work much more smoothly. Imagine Doom without even the pretense of story. It gets a lot weirder as you are just some random guy with a collection of weaponry fighting things looking like hell beasts. A lot of questions pop into mind like "wtf is going on?" However, add the simple barebones story that Doom has (you're a marine on mars and a hell gate just opened) and the action can flow a lot better. Does it make much sense? No, but there are at least straws to grasp at.
More At The Table
At the table, especially with table top RPGs, the narrative has a lot of work to do. It provides the motivation, explanation, and flow, but it also carries the enjoyment. The obstacles and pitfalls that happen are part of the story. The actions that the PCs take are part of it. Being aware of this, controlling it, and shaping it are the keys to making a good game into a great game.
Next to that, the mechanics are secondary. People very rarely are remembering that awesome "24 damage that got the kill" on the monster because of the die roll. They're remembering it because "I thought I was dead, and then BAM, a lucky shot and I won." Use the story, work the story, and you can make memorable moments that your players will talk about for years.