Konstruiren can have many meanings:

Langenscheidt: konstruieren Verb/transitiv (bauen) construct (a. Mathematik u. Gedankensystem etc.); Linguistik construe; (herstellen) create; (erfinden) fabricate

Of course, you’ll have to pick the words in English that fit the meanings in your text.

The text reads:

"Menschen, die Ebene E7 und E8 erreichen, beginnen immer mehr zu realisieren, wie sie die Welt interpretieren und inwiefern dies durch ihren persönlichen und kulturellen Hintergrund beeinflusst ist. Insofern nehmen sie die Subjektivität ihrer eigenen Sichtweise zunehmend in Betracht."

As I see it, this section of the text has to do with how they increasingly gain an understanding of the mechanisms that they use to interpret the world around them. Since construe can mean to understand (or interpret), I think "sich konstruiren" means (here) to "construe themselves" and the world around them, as Michael has translated it.

But the text goes on to talk about how they increasingly gain an awareness of how they "construct" the notion of a stable world, which would, in my opinion, be a more appropriate translation of "... der Bedeutung dessen, wie sie diese Vorstellung konstruieren, zunehmend bewusst." So perhaps you need to translate the text using both "construe" and "construct," (as Phil has suggested), i.e. something along the lines of: "Indeed, they construe themselves and the world around them as predominantly stable, but are increasingly aware of the significance of how they construct this notion."

It’s less a question of style (not repeating the word “construe” in the same sentence, for example) and more a question of reflecting the underlying meaning of the German.

Hope that helps.