Every pair of lines here follows the same pattern: a request, then a warning against the opposite. "Be mine" is followed by "Don't be hers," and "Take my hand" by "Don't leave me hanging." The speaker wants closeness, trust and understanding from someone, and fears losing that person to a rival or being misread.
The tone sounds confident at first, since the lines are short and sound like orders. But the "don'ts" give away how anxious the speaker is. Someone sure of being loved wouldn't need to say "Don't trust the wrong ones" or "Don't misunderstand me." The poem seems to ask for love while bracing for rejection.
The title, "Language of Love," fits the last pair best. Learning the speaker's language suggests that love means being understood, not just being chosen. The mood is hopeful and a little desperate, like someone who has said everything plainly and is waiting for an answer.