I often read essays aloud as one of the filters before publishing them. It helps me find awkward sentences. I discovered this trick when practicing talks, but now I do it even for essays that aren't talks.
In Roy Blount Jr.'s Alphabet Juice (under "mnemonic"), he mentions that the ancient Greeks and Romans generally read aloud, even to themselves. (We know this because Julius Ceaser didn't, and his biographers considered his reading silently, and thus more quickly, to be novel. )
I often read my writing aloud in my head. Which is to say: I imagine myself reading it aloud, at what (to me) feels like the right speed.
There was a time when I used to speed-read fiction books. (I was often impatient to know how they would end.) But I caught myself doing that and deliberately trained myself to read good prose at a more moderate pace.
One thing you really should read aloud is poetry. The art makes a lot more sense when you savor the sounds of the words. You should also read poetry fairly slowly. Make each word count. Every one of them was lovingly designed for that purpose.
I read aloud when I've finished a draft of a screenplay, to help in making the dialogue better.
Also, I read a short story of my own (just 2 pages) out in 1997 to members of a writing class, and they all laughed at the places I expected and clapped at the end. It's a nice experience.
Since it was a discovery for you, did you experiment with other tricks? In particular, did you try to simply slow down your silent reading, and found out that it didn't do the job?