The best professor I remember from my undergrad days in terms of teaching the vocation of writing gave similar advice. His was to write, read, delete, and re-write from memory. The idea being, that you'll wind up cutting what needs cut without bias that way. That clever sentence you're trying to save isn't so clever if even you can't remember it after reading it.

Writing isn't hard, it's just work like anything else. I think people have a false perception of it requiring some sort of talent, when in fact anyone can eventually hit a home run if you let them take 50 swings.

PS: the reason reading it out loud helps is that certain multiples of syllables per sentence tend to sound better. People tend to pair stressed / unstressed syllables rhythmically (iambs, using the ancient word, and hence the iambic notation mentioned in reference to Shakespeare's iambic rhymes, for instance). Recreating the same rhythmic patterns in writing makes the phrases on the page stand out as your mind voices them internally. So, always finish paragraphs with short, strong, 5 or 10 syllable phrases and you'll have people hanging on your every word and thinking you're singing the gospel to them.