Hawking said virtual particles can almost become real particles, following closed trajectories. What does that mean? Here's hawking quote from his latest book. Brief Answers to the Big Questions
"When space-time gets warped almost enough to allow travel into the past, virtual particles can almost become real particles, following closed trajectories. The density of the virtual particles, and their energy, become very large. This means that the probability of these histories is very low." What's the name of experiment in which virtual particles became real. And why does their density and energy become large? And how it is related to the probabilities (of time travel)? 
 A: I'm obviously no Hawking, but I don't want your question to be ignored, especially as you had the interest and curiosity to ask it. I'll do the best I can to answer what I think Hawking meant.
You might know the quote from Hawking regarding "A Brief History of Time", about which he said:

"Someone told me that each equation I included in the book would halve the sales."

In the book you quote, (which was completed by others), the same principle of avoiding mathematics was applied, only now it applies to physics terminology as well.
At all level of particle physics, except for popular science books, virtual particles mean a purely, totally, mathematical device for calculating the probability that real particle will interact. We can't measure any virtual particle, as we can a real particle, we can't even tell how exactly the interaction occured.
We only have the fact, from measuring the real particles, that they did undergo an interaction.
If you read this page, On and Off Shell Interactions, you will see that, as a handy shorthand mental picture, we can, in extremely loose terms, think of the interaction as involving very short lived entities, and this is what Hawking et al are referring to as virtual particles in their book.
If these imaginary particles were somehow able to live long enough, we would call them almost "on shell"...(please read the link, they do a better job than I can to explain things here). 
The above is guessing on my part, but from now on  it's  a serious handwaving, totally  amateur, stab in the dark on my part, regarding what Hawking meant.
If you could go backwards in time, or have a closed loop in spacetime, or in any way lengthen the lifetime of these entities, they would get closer to something we could measure, therefore closer to real particles.

And why does their density and energy become large?

If you are trapped in a spacetime loop, and these entities are still  being produced, by interacting with each other, then the density and energy will increase, there is nowhere for it to escape to.

And how it is related to the probabilities (of time travel)?

This question I can answer with certainty: I have absolutely no idea, sorry.
