Are trees really going to explode because of the cold? Well, no, but it's also not totally incorrect.
It's called 'frost crack,' and it's a rare occurrence in trees.
Frost crack can happen in two ways. Tree sap has a lower freezing point than water, approximately -20°F. As the sap freezes, like water, it will expand. As it expands against the wood, the pressure is enough for it to crack.
The second way is the sudden cooling of the tree, causing the bark to contract against the slower-cooling wood. This would cause the bark to crack.
What's the real risk here? Probably just some loud noises and a possible falling branch -- but we're not going to wake up to a wasteland of trees.
As for the tree, frost crack damage is normally minor enough not be too concerning, but in rare cases, the crack can leave permanent damage.