What is the meaning of this phrase? “at the expense of being wrong”

As user208959 linked to, the text is from a job posting ad:

We are looking to assemble a small elite team. We want people who
value being right, but not at the expense of being wrong.

at the expense of is an idiomatic expression in English, meaning that doing the first thing harms the second thing. For example:

She acquired power at the expense of friendships.

He argues that the tax cut will benefit the rich at the expense of the poor.

She’s focusing on math at the expense of her other subjects.

In the example you’ve provided, I agree it is a confusing example of the usage of the expression. I believe the person posting the ad is trying to say that they want the best people for the job – people who do the job right, but not so sure they are right that they are unwilling who seek advice or listen to other people if they happen to be wrong. In other words, they are looking for open minded people who, presumably, work well in a team.