A process is a data structure that an OS requires to execute a program. For a given program, we create a process in the main memory. A process consists of Data Section (consisting of global and static variables), Code / Text Section (which contains executable machine code), Heap and Stack (to support dynamic memory allocation and function calls).
So when the scheduler schedules a process, the OS takes that process's context (PCB) and puts it in RUNNING queue, just to keep track of the what process's are running.
When we say "a process is running or CPU is executing a process"
or in your words
process is given to the CPU
, what we mean is that the CPU is taking executable machine code from that process's (which is living inside the main memory) text / code section and executing it line by line.