to set a person or a claim forward so that it stands established: to commend someone, or to prove something true
The picture behind this word is of setting one thing next to another so that it stands established in plain view. When the thing set forward is a person, the word means to commend or introduce them: to put your own name and reputation behind someone so that others will receive them. This was the ordinary language of a letter of introduction in the ancient world, and Paul uses it exactly that way when he asks a church to welcome Phoebe.
When the thing set forward is a claim or a quality, the same act becomes proving or demonstrating: laying the evidence alongside the claim so that it is shown to be true. Paul says that human failure actually demonstrates how right God is, and that God proves his own love by the death of Christ while we were still sinners. Sharper, more technical words for proving something exist, such as ἀποδείκνυμι, but this word carries the same force with a warmer, more personal edge.
These are not two unrelated uses but two ends of one idea. Whether the word means commend or prove depends mostly on what is being set forward, a person or a point, and in Paul the two often shade together: the commendation he trusts is the kind that is publicly demonstrated by action and endurance rather than merely claimed. That is why he can insist that the only self-commendation worth anything is the sort the Lord himself confirms.