As stated, the entire play is poetry. It just doesn't all rhyme (though a great big portion of it does). It's the meter that makes it poetry. The whole thing is written in iambic pentameter, meaning ten syllables per line, five of them stressed and five unstressed. So EVERYONE is speaking in poetry.
Now, if you are talking about the SONNET Romeo and Juliet share when they first speak to one another, the reason for that is that Shakespeare obviously wanted the entire audience to understand immediately that Romeo and Juliet had already fallen in love. Sonnets were considered the highest artistic expression of love. They were also wildly popular, and a recognizable form. The Elizabethan audience, even the illiterate ones, would have known they were hearing a sonnet the same way we would know if we were hearing a knock-knock joke.
It's very romantic. Romeo speaks the first four lines of the sonnet, Juliet speaks the next four, then they alternate; and their first kiss occurs halfway through the last line. The audience would have been thinking "wow!..."