One example of compiler intrinsic functions is the floating point math functions that are part of the standard C math library. On CPU architectures with on-chip support for floating point trigonometric functions like sine and cosine, the compiler might replace the function calls with a series of instructions that achieve the same result but using the available CPU instructions instead.
Another example is replacement of calls to C low-level memory copying functions such as memcpy and memmove. These might be implemented directly using SIMD instructions if available, copying the blocks of memory many times faster.
Note that "... an intrinsic function is a function available for use in a given programming language whose implementation is handled specially by the compiler." This is in contrast to operators like addition and subtraction; the compiler would be expected to implement floating point arithmetic operators even without an FPU since the C language specification demands it. Intrinsic functions go further and implement things that are properly part of the C library of hosted implementations.