If you step into a puddle on a cold day, your feet will probably get wet and once they do, they'll be cold until they dry off.
Before stepping into that hypothetical puddle, your feet were probably warm because you were wearing a pair of socks and a pair of shoes. Dry socks keep your feet warm by trapping a pocket of air between your feet and your shoes.
Why So Cold?
Air is a great insulator and your feet need insulation in the cold because without it the heat of your feet will quickly transfer to the cold of the ground. A pocket of air keeps this heat loss to a minimum. When you step into a puddle, the pocket of air created by your socks becomes a pocket of water.
Unlike air, water is a very poor insulator and the warmth of your feet quickly travels through the water and is lost on the ground.
Once your feet lose heat and your body begins getting colder, your circulatory system starts decreasing the amount of blood it sends to your hands and feet. With less warm blood flowing to your extremities, it will take your cold, wet feet a long time to get warm again.
The moral of this story is that your mother was right! When it's wet and cold outside, it's best to wear a good pair of socks, a pair of well-insulated boots, and if you want to keep warm, be sure to avoid the puddles.