When writing a story, if you are similar to myself, it's a matter of getting what's in your head down on the screen. Annoyingly, what's in your head can come out faster than you can type. An obvious reason to read it back later. To tidy up mistakes and punctuation.
When you read back through what you have written you need to keep an eye out for two things: word order, and excess words. Let's look at excess words first, it's probably easier. There are words you throw in all the time which are unnecessary. Once you are aware of this you will start to catch what you do subconsciously, and you will be able to remove your favourite foibles! As an example, I have an annoying habit of writing "Just." Just this and just that. If I didn't go through removing this repetitive and mostly meaningless word, it would appear near the top of the words most often used. Another such word is "really." "He really meant it!" "He really was good at..." etc., etc.
Let's look at a whole sentence:
But the real reason why Steven was so angry was that today was the day when his friend Nick was supposed to come over to play video games and, for the first time, stay the night.
The reason Steven was angry today was because his friend Nick was supposed to come over to play video games, and for the first time, stay the night.
I've taken out nine words which pad out the sentence and put in one word that arguably need not be added, but joins the two halves of that sentence. Read the two sentences again. Does the second read better? Do you need "But" at the start? The "real" reason, is there another pretend reason? The reason why, what does that mean? It's simply the reason. So angry, "so" doesn't qualify angry. It would work in dialogue, but in narrative, one is angry, very angry, or extremely angry. All these are degrees of anger, "so" is not. The word only works combined with a visual expression, hands apart, "it was so big" indicating visually the size. The day when, it's simply the day, why "when?" What does that word add? The day when it happened, the day it happened.
I could give a whole list of padding words, but I doubt neither you or I would successfully eliminate them all. The point is to make ourselves aware. Not to cut the spontaneity of writing, but to take the time to step back and read over. More concise sentences make for better reading.
Word order is more tricky. Try to think about it like this: you are tapping out your story as quickly as you can get the thoughts down. You write a sentence and think it needs qualifying. I've read, "He softly spoke..." Doesn't sound right. "He spoke softly..." "I went to catch him up, quickly." You can almost see the thought process of the author in that sentence. He's writing an action scene and wants to emphasise the speed of movement. When you read it a second time, you need to ask yourself, should it stand? Would it be better changed around? "I went quickly, to catch him up." Or better still, "I raced after him." You aren't necessarily going to come up with the best narrative first time around. Dialogue, maybe, but narrative needs a pause and reconsidering.
Immediacy is the watchword of today's modern author. You might not agree, you are more than entitled to your own point of view, your own style. Consider, most novels are written in the past tense, but read as being immediate. More and more authors chose the present tense. I'm not discussing here the pros and cons, simply pointing towards the trend for immediacy. Which means putting the reader in the thick of the action.
Take an example which is representative of how a lot of us write: "When you're sitting in a dark and hot and increasingly smelly bathroom you have a lot of time to think, and you begin to wonder why the Hell your stupid parents had decided..." Had is a word both unnecessary and which puts the action back a step. It is not immediate. Using "When you're.. " evokes the reader to imagine a memory of a past situation, it's not immediate. "Sitting in a dark..." puts the reader right there in the action. You're telling a tale, but the better you engage the reader, the better their experience. Putting them right there in that bathroom, is better perhaps than evoking a memory of what it would be like.
If you want the reader in the action, you might have written: "Sitting in the dark, hot, and increasingly smelly bathroom, with a lot of time to think. Why the Hell did your stupid parents decide..." I'm not telling you this is better than the original. Simply it is something to consider when you read through your story.
We all make typos. It's hard to correct our own. Use of Spell Check gets us a good way down the road, but it fails to pick up worms that have been swapped with other words. Spill Chuck is fine as far as it goes, but the marque won eyeball (ok, I've stopped now) is the best tool ever.
A second pair of eyes will pick up more than just yours.
The other things are important, too. While your dialogue may be eccentric in all manner of ways, your prose must be as perfect as you can make it. We accept dialogue for what it is, but we go back to English lessons when we read prose.
Don't use constructs you don't understand. No knowledge of how to use a semicolon? Don't use them, then!
When we write we sometimes forget continuity. It can be as simple as the colour of a pair of socks, or as awkward as impossible dates and ages
I'm sure J K Rowling was very pleased with Harry Potter and the Incredibly Long Camping Trip, but not many readers were impressed. Rowling ought to have been edited down heavily for material that added no value to the plot, but wasn't.
Writing in our genre means that people are not invested in reading our stories. They haven't paid for a book. Anything that gets boring gets ignored. And that means we need to edit our work and consider a relationship with an editor, who will spot the crap and get us in the right frame of mind to lose it.