This is a way of using lots of different tunings without having to relearn all your chord shapes with different names. Tune to the given tuning, play the shape indicated, and everything else takes care of itself.
The result is that in terms of absolute pitch you'll actually be playing a different chord - you're transposing using a different tuning.
I'm not sure about your particular book, but generally the C (GCEA) tuning is taken as the reference. So they say to tune to ADF#B. If the song says to play F, play your regular F shape (2,0,1,0), but in terms of actual pitch it will come out as a G - because you've tuned the whole uke up one whole tone!
So you think "play F", your fingers "play F", but the actual sound is "G".
The point of this is to play it the way George himself would have. I think he generally had a few ukes on stage to handle the different tunings. If you don't want to retune all the time you can ignore this, and just play the tunes as written without the retuning. You may end up playing the tune a little lower than George, but that's no harm unless you need to play along to his recordings.
Another option is to use a capo instead - for ADF#B capo at the second fret. For BbEbGC, capo at the third fret.