As an example of "deadly serious and a little distant," check out when Bigwig gets caught in a snare and he's not moving:
Blackberry pressed his nose against Bigwig's head. As he nuzzled him gently the head rolled sideways and back again.
"Bigwig," said Blackberry in his ear, "the peg's out."
There was no response. Bigwig lay still as before. A great fly settled on one of his ears. Blackberry thrust at it angrily and it flew up, buzzing, into the sunshine. (17.64-6)
Now, this is a serious moment, where one of our main characters may be dead. Notice what this section doesn't do: it doesn't make a joke about Bigwig's death; but it also doesn't go on and on about what a tragedy this is. There are no jokes when it's a matter of life-and-death, but it doesn't tell us how to feel. We're not being manipulated into growing weepy and sentimental. It just shows us the situation and let's us come to our own emotional conclusion. That's what we mean by distant.
Also, check out how detailed that scene is, how the narrator can slow down the action to tell us basically the same thing twice: "There was no response" pretty much means the same as "Bigwig lay still as before." But by repeating the same info, the narrator slows down the action so that we can insert our own feelings into the story.
So when we read that Bigwig might be dead, the narrator doesn't tell us that this is a big freaking tragedy, but slows down the action so that we can feel how tragic this circumstance is. Their friend is dead, so it's like time is standing still. The narrator doesn't bombard us with feelings, so we can have our own emotional response, which is, for those of us who have hearts, horrific.
But notice also how close the narration stays to the rabbits. Even though it's an omniscient narrator, we mostly hear what the rabbits are feeling or doing. We're not going to get, for instance, to see how disappointed the farmer is that he doesn't get to eat a delicious rabbit. And that's why we say the tone is only a little distant.