If polyethylene glycol (PEG) is hygroscopic, how does it protect wood?
The reason why PEG is suitable for this purpose has many reasons including the wood chemistry and degradation of wood in water.

Degraded wood usually very characteristic chemical losses. The cellulose of the wood is usually lost due to bacterial action and/or due to the action of the water itself. Over long period of time wood looses water-soluble components like starch and complex sugar like molecules to water. What remains in there is the water-insoluble component in the cell walls and wood tissues primarily consisting of lignin. 

If the wood is in water, lignin provides structural stability to the wood. But wood becomes spongy and easy to break. If this wooden piece is exposed to air, water evaporated and wood becomes dry and shrinks. Most of the spaces between lignin layers is nor air, which makes it easy to break. 

Here, comes in PEG. PEG is remarkable because it can dissolve in water. 
PEG solution has very high osmotic pressure and this is one of the reason why PEG solution treatment works. PEG seeps in the tissues and provides the required osmotic pressure to keep lignin (and wood tissue) stable. This prevents the wood from expanding and shrinkage. While treatment, it is ensured that PEG penetrates the wood tissue which takes considerable times, and water is removed from the wood. In the end a large part of the original water in wood is replaced by PEG in the tissues providing structural stability.

Coming to your point on the hygroscopic property (also known as deliquescence). Higher molecular weight PEG has lower hygroscopic property because of more alkyl chained part. So it depends on the type of PEG used. But a balance has to be made since, lower MW chains are easier to penetrate tissue.

In case the relative humidity changes, for example increases, the effect due to absorption due to PEG inside tissue is negligible. Only the PEG which is outside tissues (excess, which was not removed during PEG treatment) might cause additional increase in weight. The removal of excess PEG is usually taken care of by experts. 

Still however, it is the norm that >80 % relative humidity is not good for PEG treated wood.

Reference


D.W. Grattan, "A Practical Comparative Study of Treatments for
Waterlogged Wood, Part II, The Effect of Humidity on Treated Wood",
(Ottawa: ICOM Waterlogged Wood Working Group, 1981), pp. 243-252.
Taduteru Nishiura and Setsuo Imazu, "Experimental study of the
dimensional change of highly degraded waterloged wood according to
ambient humidity after preservation treatement"Bremerhaven: ICOM
Waterlogged Wood Working Group, 1990) pp. 107-118.