An important part of experiencing and loving flowers is knowing their names, stories, and meanings. Most of us are familiar with Shakespeare's line in Romeo and Juliet, that "A rose by any other name would smell as sweet." This is true and flowers are for everyone, even those with zero interest in horticulture.
Botanical Latin is hard to love, but it helps make sure you get the plant you want, and it often gives clues to size, color, fragrance, texture, traditional uses, and place of origin. It's also an important asset when reading garden writing, since writing can range from all common to all Latin nomenclature. I truly hated Latin when I was in school (barely passed it all three years), but now I'm working to learn late 19th and early 20th century botanic names (which have often changed due to new research and DNA testing), so I can read the classics of garden writing without frequently Googling antiquated terms.
Common names are useful as a translation for people who are not plant people, but can also offer insights about a plant's traditional uses, flavor, scent, physical qualities, bloom time, and origin. For some plants the common name is a portion of their botanical name (Boring!!!), and others can have dozens of common names, based on their geographic reach and cultural importance.
The story of a plant includes:
1) its official Western history (geographic origin, who "discovered" it, which plant breeder created the cultivar);
2) its indigenous history (the plant's uses and meanings for the people who have been using/cultivating it for thousands of years);
3) its unofficial history - your history ("this is my Aunt Maryellen's rose", "my neighbor Bob gave me cuttings", "I found this heirloom pumpkin through Seed Savers", "we were married under that oak", etc.); and
4) its "Victorian Language of Flowers" meaning (the red rose for love, and bay laurel for success or triumph are probably the best known, but there are several hundred plants in the Anglo-American tradition that have one or more meanings attached - whether it's a long or short list, it's safe to assume that every culture and civilization has its own "language of flowers" and they are all equally valid as a form of human expression).
As Shakespeare made clear, you don't need to know any of this background to love flowers. At the same time, I would encourage everyone to take a deeper interest in their favorite flower or favorite genus (e.g. daffodils, lilac, or dahlias). Few of us want to know everything about the vast subjects of wine, beer, or spirits, but I think most of us enjoy knowing the history of our favorite winery, brewery or distillery. Your commitment to being a plant nerd is a personal decision, but at least dip your toe in the water!