A flourishing business or garden means that it's doing very well and growing strongly.
You'll often hear people talk about something flourishing when they mean that it's successful and thriving. For example, a new restaurant might say they're "flourishing" after a positive review from a famous food critic, which means their business is booming and lots of customers are going there to eat. On the other hand, if someone talks about a garden that's flourishing, you can imagine it as being full of healthy plants and colorful flowers that grow well in that environment.
How common is "flourishing"?
Word flourishing is considered rare in modern English. It has a balanced usage among all categories: speech, fiction, newspapers and academic texts.
Definitions
verb
To thrive or grow well.
Example: The barley flourished in the warm weather.
To prosper or fare well.
Example: The cooperation flourished as the customers rushed in the business.
To be in a period of greatest influence.
Example: His writing flourished before the war.
To develop; to make thrive; to expand.
To make bold, sweeping movements with.
Example: They flourished the banner as they stormed the palace.
To make bold and sweeping, fanciful, or wanton movements, by way of ornament, parade, bravado, etc.; to play with fantastic and irregular motion.
To use florid language; to indulge in rhetorical figures and lofty expressions.
To make ornamental strokes with the pen; to write graceful, decorative figures.
To adorn with beautiful figures or rhetoric; to ornament with anything showy; to embellish.
To execute an irregular or fanciful strain of music, by way of ornament or prelude.