Almost every line of "needy" begins with "I want," and that repetition is the whole shape of the poem. The speaker is picturing a relationship with someone they long to be close to, and they pile up wishes one after another: to be known, to be held, to trust and be trusted, to come home to someone, to share small silly moments. Some wishes are big, like wanting to love again, and some are tiny, like wanting to have "stupid cute couple moments."
Underneath the longing there's hurt. The speaker wants to hear they're "not broken" and to be seen as "whole," which suggests they feel damaged, perhaps by past love. Wanting to speak "without fear" hints at how hard honesty feels right now.
The mood is tender, vulnerable and a little anxious. The title admits the speaker thinks they're asking for a lot, yet the final line, hoping the other person wants these things too, shows what they really need: to be wanted back.