What are the benefits of studying poetry?

This is a great question! There are definitely many benefits to studying poetry for many groups of people. Here are some:
1) Language learners can greatly benefit from a concentrated immersion in poetry. As poetry emphasizes rhythm and stress patterns, many ESL (English as a Second Language) learners can hone their pronunciation skills by reading poems aloud. Reading-aloud sessions can provide the requisite practice to increase the confidence of language learners.
Additionally, in studying poetry, language learners and other students can begin to appreciate and discover the history behind the poems they read. For example, Shakespearean poems provide a wide context from which to discuss relevant topics such as gender relations, human sexuality, and cultural norms during the Elizabethan era.
The Elizabethan era, as we all know, heralded the golden age of the Renaissance as well as the new age of English exploration and expansion. Therefore, an appreciation of the historical context of the poems can lead to an enriched and stimulating experience in the classroom. For example, students can compare and contrast the differences in gender relations between the modern age and the Elizabethan age.
2) There are emotional and intellectual benefits to studying poetry. For example, a poem may lend itself to many different interpretations. A discussion of these interpretations can help students hone their logic and reasoning skills.
Additionally, an appreciation for the metaphorical or figurative language used in poems can inspire students to view familiar objects in imaginative new ways. Students can increase their emotional and verbal intelligence by immersing themselves in the spirit of the poems they study; thus, the practice of exploring the possible, hidden meanings behind different poems can inspire an openness to new ideas. This approach can only increase a student's sensitivity to the language of poems, a sensitivity that may very well be carried into his/her own writing. Thus, studying poetry can conceivably improve a student's diction immeasurably.
3) In the business world, reading poetry in one's spare time can do one of three things. It can aid in inspiring creativity, improving empathy, and simplifying complexity. For example, reading a complex poem can be frustrating as well as rewarding. Making sense of a poem can be a consuming process, but it can also hone our ability to simplify the kind of complexities that intimidate others. Sometimes, this ability can be a good thing, especially when deadlines loom and solutions are needed.
Poetry also teaches one how to empathize with others. Through poetry, we can begin to appreciate the different perspectives of others. This appreciation can lead to constructive dialogue and perhaps, understanding between different social groups. Last, but not least, poetry inspires the kind of creativity that is unique and often revolutionary, especially if applied to the business world.

Clare Morgan, in her book What Poetry Brings to Business, cites a study showing that poems caused readers to generate nearly twice as many alternative meanings as “stories,” and poetry readers further developed greater “self-monitoring” strategies that enhanced the efficacy of their thinking processes. These creative capabilities can help executives keep their organizations entrepreneurial, draw imaginative solutions, and navigate disruptive environments where data alone are insufficient to make progress. (from The Benefits of Poetry for Professionals by John Coleman).

A great article to read: Why Teaching Poetry is So Important.
Hope this helps!
https://hbr.org/2012/11/the-benefits-of-poetry-for-pro

https://www.theclassroom.com/benefits-reading-poems-11971.html