In just one sentence, the speaker admits something they claim they can't admit. That's the joke, and the sting, of the poem: by saying "I am too arrogant to admit I am shy," they have already admitted it. The confession slips out even as the speaker pretends to hold it back.
The title, "Pride," points to the main tension. Shyness feels like weakness, so the speaker covers it with a proud, distant attitude. Someone who seems arrogant or aloof might really be afraid of being seen. The word "too" matters here, because it suggests pride is stronger than honesty, strong enough to block the truth, though not strong enough to hide it from us.
The mood is wry and a little vulnerable. There's humor in the neat twist, but also some sadness, since the speaker seems trapped by their own defense. It reads like a quiet moment of self-awareness, said almost under the breath.