Answer:
They're made from super-absorbent polymers (classified as hydrogels) that can absorb and retain extremely large amounts of a liquid relative to their own mass, thus increasing both size and volume. (This is the same type of material found in disposable baby diapers, adult protective underwear, and feminine menstrual pads.)
The total absorbency and swelling capacity of SAPs are controlled in part by the type and degree of cross-linking to the polymer, but it's also determined greatly by the ionic concentration of the absorbed aqueous solution. Deionized and distilled water can absorb up to ten times more mass than non-distilled, partially salinated water (i.e. tap water). This means that, if a toy can grow up to up to 5 times its own size in tap water, it might grow up to 50 times its size in deionized distilled water.
So if you use deionized distilled water, you really can grow your own life-size girlfriend or pirate ship.