When people describe gender as a "social construct," they're saying that qualities we tend to consider "innate" are actually a product of the way genders are differently socialized in society.
Think, for example, about the stereotype that women are inherently more emotional than men. Those who believe gender is a construct would say this has no basis in human biology. Instead, they'd see it as a function of the way women are raised. Though there are always exceptions to social trends, it's true that most young girls are encouraged to embody certain characteristics that have historically been considered feminine—characteristics like softness, vulnerability, or nurturing. Over the course of her life, a woman might internalize these expectations and ultimately feel that she doesn't get to assert herself, take initiative in her own life, or focus on her own needs instead of the needs of others.
Conversely, think about the stereotype that men are more logical than women. This doesn't just create a dynamic where men are placed above women in the intellectual hierarchy, it also creates an environment where men are not permitted to openly acknowledge their own emotions. This places an immense and unrealistic expectation of stoicism on men, even as it appears to socially "privilege" them. Often, when men need mental or emotional help, they face a stigma even greater than that faced by women.
When gender is viewed as a social construct, this causal paradigm appears extremely harmful. If a person is taught that correct performance of their gender requires specific behaviors and prohibits others, those "rules" often become deeply ingrained and very hard to extricate from one's true identity. When one's identity doesn't comply, they can often face inner conflict and outer stigmatization as a result.