A construct is a theoretical concept, like intelligence or extraversion, that can't be observed directly but is inferred from patterns in observable behavior.
You can't point to "extraversion" the way you can point to height. Nobody has ever seen a trait under a microscope. What you can observe is a person talking more at parties, seeking out social situations, and feeling energized by group settings — and from that consistent pattern, psychologists infer an underlying construct that explains it. The construct is the theoretical story; the behaviors are the evidence for it.
This is why psychological measurement is harder than physical measurement. A ruler measures length directly. A personality test measures a construct only indirectly, through a set of questions assumed to reflect it. That gap is exactly what concepts like validity and the nomological network exist to address: since you can't check a construct's measurement against direct observation, you instead check whether it behaves the way theory predicts across many related measures.
Not every construct turns out to be useful or real in the way it was first proposed — psychology has a history of constructs that didn't hold up under scrutiny. That's a feature of the process, not a flaw in it.