I can sight read for winds and voice, but not for Piano, despite being able to play Piano relatively well. Thus, this explanation will probably apply mostly to single-voice musicians.
1. Find Do. This is the first note of the Major scale in the key which you are playing. Find it on the page, and burn it into your mind; everything you're about to do centers on the location of Do. In the key of C (white keys on a piano) Do is right in the middle of the grand staves, keeping things simple. If you know where Do is, you don't have to care about the key signature for like 97% of the notes you're going to read, and you can make educated guesses about what the sharps and flats do by using your musical intuition. Once you've found Do on the page, figure out where Do is musically, and keep it in your head. Make sure that when you're singing or playing any other note, you know where you are relative to Do, and can jump back there without losing your place.
2. Learn your intervals, and learn Solfege. If you're a musician, you already know these things in principal, even if you don't know the words or the terminology. In particular, you should be able to "hear" the distance between Do-Mi-Sol without too much difficulty, because you hear those distances in music a lot.
3. When you're sight reading the first note in a passage, find the note, then step it off visually to the nearest Do. If you're good, you can also step it off to the nearest Sol. I usually root myself on Do and Sol both in a given passage, as it gives me less visual separation between any note I need to find in a hurry.
4. For every note after that first note, pay mind to the spacing on the page:
4a. Moving up and down the scale should be easy; you're a musician, you already know what a scale sounds like, just sing along.
4b. Moving up or down from one line to the next line is a third. Most often you'll hear these relative to Do-Mi-So-Ti in major keys, and La-Do-Mi in minor keys. You'll hear these a lot, so get used to reading them quickly.
4c. Get a good feel for common jumps while you're at it. As a Bass singer, I got really good at reading jumps between Do-Fa and Do-So, because those chord progressions exist in everything, and their sole purpose in life is to make bass singers bored out of their mind. You're a human, in addition to a musician, so you'll pick up on the patterns in any given composition and have the common tricks down before you even realize what your brain is doing. Don't question that, let your brain be awesome all on its own.
That sounds like a lot of steps, until you realize that you're performing most of them already to make sense of any piece of music that you play. Don't worry about the names or the terminology, you can use numbers (1-3-5) if that's easier, or just "hear" it if you don't feel like you need an aide. Remember, the goal is to drop the aide and be able to just perform these steps on your own eventually.
I'm analytical in nature, so I'm always thinking about these things. I suspect the reason most musicians can't describe this process very well is because the process of analyzing our craft detracts from the quality of its output. The more you can commit to "muscle memory" as a musician, the more you can pull your analytical mind out of the process of a performance, the easier it becomes. Your body's built in reflexes are faster than your analytical brain will ever be, and good musicians tap into that biological strength through loads of practice.