What is the significance of "microcontroller based on microprocessor"?

ARM is the CPU core, which can be used to implement a microprocessor or a microcontroller.

The full sentence you referenced is:

The STM32 family of 32‑bit Flash microcontrollers based on the ARM® Cortex®‑M processor is designed to offer new degrees of freedom to MCU users.

Clearly this is largely content-free marketing babble. Don't pay much attention to it. It actually never says "microprocessor". It refers to the ARM core as a "processor", which isn't strictly right either. It's just a core, which can be used to implement various kinds of processors.

The core is more like the engine of a car. You license the design from ARM, but can heavily configure it to your needs, and putting the chassis and wheels around it is your job. You can make the result a sports car, a pickup truck, or various other types of vehicles. The marketing babble above is like saying "We've based this pickup truck on a sports car". No, they haven't. They've base the pickup truck on the same basic engine technology others have used to make a sports car with.

Again though, the important point is that this is all marketing babble. There is nothing useful to see here. Move along everyone.

A microprocessor is effectively a computer without any peripherals (I/O, ADC, timers, etc). A microcontroller is a processor with those peripherals connected and all combined into one package. Therefore, a microcontroller is based on a microprocessor, and adds peripherals to it.