I don't think he is saying, "good explanations are hard to vary while preserving their predictions".

As described above the statement "Everyone just acts in his own interest" very easily preserves its predictive power in a multitude of situations. Indeed, the problem with it is that the statement preserves its predictive power in too many situations! The explanation is consistent with just about whatever happens, so one cannot design a test that makes one believe that the statement is certainly false. So it is too easy to vary and hence a bad explanation.

Evolution does not increase a species' implicit knowledge of the niche by replicating genes. Mutation (evolution's conjectures) creates potential new knowledge of the niche. Selection decreases the "false" implicit conjectures of mutations and previous genetic models of the niche.

So induction does not increase the implicit knowledge of gene sequences.
Trial (mutation) and error (falsification) of implicit theories does. This is the process that the critical rationalist says happens but more efficiently with humans.

He does not mean "lacking unnecessary details". For example the statements "Everyone just acts in his own interest" or "Everyone is really an altruist" are simple and lack unnecessary details, explain quite a lot, and are consistent with Occam's razor. But by Deutsch's criteria they are bad explanation because they are too easy to vary. For example, someone who believes in the self-interest theory could say, "John gave to charity because he would have felt guilty otherwise. So he really was selfish" .

We see that it is easy to change the theory that everyone is selfish to accomodate the case of someone who seems altruistic.

Or someone who believes in the altruist theory could say about John murdering Harry, "Well then, Harry must have been very unhappy."

The altruist theory and the selfishness theory are simple and explanatory in their own way, but too easy to vary. Similarly the idea that that sexism, feminism, capitalism, communism, parental coercion, environmental disregard, etc. cause unhappiness or mental illness or some other broad conclusion are equally meaningless. These explanations are bad because they can be varied to explain ANYTHING.

In contrast, theories that are difficult to vary go out on a limb. They are bold conjectures that explain a lot but even one small counterexample easily invalidates the whole thing. A good theory can not easily be changed to "take into account" the aberration. For example, Einstein's theory of gravitation is a good explanation because it explains a lot, it makes counterintuitive predictions, and even one repeatable counterexample invalidates the whole thing. It can't be easily changed to accommodate something else without invalidating everything else about it.

Theories that are hard to vary remain constant over time. They are more true and therefore more timeless. Invariable theories possess more verisimilitude ("truth-likeness" to use Popper's term).

Like the very best possible theory, truth also cannot be varied. It is completely timeless. It was, is, and always will be true, without any change. That is Deutsch's point.