My four-year-old daughter asked me why paper tends to fall apart when wet, and I wasn't sure. I speculated that the water lubricates the paper fibers so that they can untangle and separate more easily, but I really wasn't sure.
Forever_a_Newcomer is on the right lines, but it's not like water dissolving salt.
Paper is mostly made from cellulose fibres (depending on the type there may also be filers and glazes like clay). Cellulose molecules bristle with hydroxyl (OH) groups, and these form hydrogen bonds with each other. It's these hydrogen bonds that make the individual fibres stiff, and also hold the fibres together.
Water is also full of OH bonds, obviously since it's H$_2$O, and the water molecules form hydrogen bonds with the hydroxyl groups on the cellulose, which breaks the hydrogen bonds that cellulose molecules form with each other. There are two results from this: firstly the cellulose fibes in the paper become floppy, because their internal hydrogen bonds are broken, and secondly the fibres separate from each other more easily. The combination of these two effects makes paper easier to tear apart when wet.
Most organic materials show similar behaviour. For example cotton is also easier to tear when wet (cotton is also made mostly from cellulose). Also hair becomes floppier and more easily damaged when wet, though the effect is less pronounced because hair contains fewer hydrogen bonds than cellulose fibres.
5$\begingroup$ This answer has changed my life. I now see hydrogen bonds everywhere and understand a great variety of phenomena that were mysterious to me before. For example, why does cereal get soggy after your pour the milk over it. Thanks. $\endgroup$ Dec 1, 2013 at 22:23
2$\begingroup$ Could you please look into this question where user Curious one has posted a link refuting the above mentioned-physics.stackexchange.com/questions/255230/…. Thanks. $\endgroup$– AbhinavMay 12, 2016 at 16:06
$\begingroup$ Hi John, I found an image of cellulose fibers showing the hydrogen bonds in between them which I believe would add to the quality of the post. Please let me know if you agree and then you or I can make the edit. Here is the suggested edit. $\endgroup$– pentaneMay 12, 2016 at 16:27
1$\begingroup$ Also explains why paper can be molded into a shape when wet, and then becomes more elastic (i.e., tends to hold that shape) when dry. $\endgroup$ Sep 16, 2016 at 13:52
You are close to mark, it's a kind lubrication.
Cohesive force between the cellulose fibers is of electrostatic origin. As water molecules, which are small dipoles, penetrate between the fibers they align with electric field. This effectively shields them of each other, greatly diminishing the force. Thus the easy untangling of the fibers (what we see as ripping).
It's similar to how water dissolves salt, here's an animation for your Daughter: http://www.youtube.com/watch?v=xdedxfhcpWo