I took a lot of this advice with me on a pretty darn hilly ride last week, along with the advice I got about being scared on the down hills and it really helped! I used to attack the hills with everything I had and never make it up the big ones.
I took the "let the climb come to me" philosophy, and just kicked down to low gear from the start and only looked up to scan every few seconds but kept my eyes closer to right in front of me and it really worked! Even the cue sheets I had for the ride warned about some killer hills and I figured I'd not make it up them but I never got off my bike- it felt awesome!
The downhills were better too! I relaxed and let myself tread on the edge of fear and while I was slower than most on the hills, I was faster than my norm- I had the benefit of some wide well paved rural roads with clear views to the end and that gave me a big safety feeling.