The speaker has just had an uncomfortable realization: the person they adore is not theirs alone. Others are chasing the same "wild precious heart," and the speaker admits to having forgotten this, as if they had been lost in their own feelings and ignored the competition. The repeated line "I forgot that I'm not the only one" sounds like someone talking themselves back to reality, almost wincing as they say it twice.
The image of a heart "pricked by a thousand darts" is a playful nod to Cupid's arrows, but it also hurts. It suggests that everyone in this crowd of admirers is suffering in the same way, and that includes the speaker. The adjectives "wild" and "precious" hint that the loved one is both hard to catch and worth the trouble.
The mood is wistful and a little anxious, with a touch of self-mockery. The speaker feels hopeful and insecure at once, caught between longing and the fear of being one of many. The title, "Hopelessly Inlove," fits that mix well.