"Front page news" is a synonym for the important or notable story. It's the news that's important enough to put on the cover of the newspaper.
So, page 14 news would be, in comparison, very unimportant, or not at all noteworthy, novel or interesting.
The irony is that a massive attack on the Pentagon would be expected to be all of those. Unless of course, it is happening often. Or perhaps if the Pentagon had ceased to be seen as important.
Edit: I was half-asleep when I originally wrote this, and hadn't spotted the fact that the item in the Onion was a contemporaneous report on 9/11. So the irony is that so many massive attacks have happened that an attack on the Pentagon is being "buried" on page 14.
The joke hinges on the phrase
in my pajamas
In the first line is ambiguous, it can be read as
One morning, in my pajamas
meaning you are in your pajamas, or
an elephant in my pajamas
meaning the elephant is in your pajamas
Of course an elephant would never wear your pajamas, so the listener will naturally assume you are wearing the pajamas. However, the following line
How he got into my pajamas I'll never know.
Confirms the fact that the elephant was wearing the pajamas, which in itself is a funny visualisation, but on top of that to shoot an elephant wearing pajamas has never been heard of before.
The joke is based on misdirection, where the listener thinks one thing, and the teller says another
A similar joke is
When I was born, they threw away the mold.
Well, some of it grew back...
Being a play on the word mold, meaning either a form or a furry growth on dead things.
Best Answer
The first one is a play on the phrase 'I'm going bananas' to mean going a bit crazy. (Sounds a bit like a Tim Vine one-liner this). It is meant to make you think they are going crazy when you read the first three words, but then when you read the rest, you realise you misunderstood (due to the lack of punctuation) and that the person is actually saying to their bananas, that they are going.
This is a joke best delivered verbally. If done with the right timing, (pausing after the first bananas) it's makes the audience think they are saying that they are going crazy, then when you finish the sentence it's clear that you're not - you fooled them into thinking you were saying one thing, but said another.
In the second one, a solid refers to both a 'favour' and 'poo'.
When you eat a lot of grapes, you tend to have softer poos, hence why it is difficult to do a 'solid' and would be annoying (and more difficult) if a friend asked you to do them a solid.
Here the person telling this is purposefully misconstruing what their friend is asking them to do.