What operating system manages resources in ARM cortex M4 based development board?
Usually those small processor boards do not have an operating system in the classical sense, they may have a bootloader or an RTOS (or both). The RTOS where present is often built into whatever codebase you write rather then being a separate program. It is common to not bother with an operating system at all.

ST have a HAL that they ship, opinions on bothering with it vary. 

Common free choices for an RTOS are things like ChibiOS and FreeRTOS (Check the licensing works for you in both cases), with things like VxWorks and QNX as commercial possibilities.

Note that the common usage of these boards has any 'operating system' be closer to a library of routines that do complicated things like thread scheduling and management rather then being an operating system in the sense of a program loader that provides resource management from a separate memory space.