Why didn't they all explode at the end of The Abyss?

The novelization by Orson Scott Card actually explains this:

Card wrote the aliens as a colonizing species which preferentially sought high-pressure deepwater worlds to build their ships as they traveled further into the galaxy.

Their knowledge of neuroanatomy and nanoscale manipulation of biochemistry was responsible for many of the deus ex machina aspects of the film;

• an NTI saved a diver's life after a breathing mixture accident,

• prevented permanent brain damage during Bud's 2 mile dive, which allowed him to properly disarm the warhead,

• a number of NTIs microscopically infiltrated the crew upon their rise to the surface to prevent decompression sickness.

(NTI = non-terrestrial intelligence)

It was James Cameron himself who wanted Orson Scott Card to write the novelization, because he wanted it to be

not just a “book of the script,” but a real science fiction novel.

Orson Scott Card was given the opportunity to visit the set, talk to the people involved in the production, and was even able to watch a rough cut of the film instead of just working off the script alone.

Orson Scott Card:

• I also had a few chances to work with (producer) Gale Anne Hurd, who was incredibly supportive during some key decisions, most notably the need for me to completely explain the NTIs. As a science fiction writer, I knew that this would be essential to making the novel a good novel by sci-fi standards.

  So Mr. Cameron’s desire to leave them quite mysterious in the film could not be carried over into the novel; on the contrary, the whole point of a novelization is to be able to tell the readers things that can’t be shown in the film.

You quoted what I was going to reference in my answer already! This seems to me like a perfect example of Lampshade Hanging which is :

the writers' trick of dealing with any element of the story that threatens the audience's Willing Suspension of Disbelief, whether a very implausible plot development, or a particularly blatant use of a trope, by calling attention to it and simply moving on.

As to what the aliens could have done they could have beamed all the nitrogen out of the humans (and mouse!) They would also have to beam out some of the other pressurized gasses. Probably a little bit at a time as they brought everyone up. They did show themselves to have a higher level of technology than we have (example the control of water when providing a room for Virgil to breath in).

I don't know if they used "beaming" technology, and have nothing to back that up, but it is possible which is what the question asks. :)