When you write C code like "a = b * c" compiler will convert that to multiply instruction. There can be of course additional instructions to move values from stack to registers and back, but the main thing is * symbol means to use multiply instruction.
But CPU has much much more instructions: http://ref.x86asm.net/geek-abc.html
There simply isn't a way to write them in C using special symbols (like + for addition, or * for multiply). So compiler or CPU vendors came up with idea of intrinsics. They look like a function call. But compiler recognizes these fake function calls, and instead of generating real CALL instruction they generate specific CPU instruction for each intrinsic. So simply this is a way how to access all the complex CPU instruction set from C code without writing assembly code.
There are bunch of advantages to writing directly assembly:
1) compiler can allocate registers for you. You don't need to worry what should be in register 0, and what should be in register 1.
2) compiler can rearrange instructions so they are better scheduled in target CPU. If you write assembly code compiler won't change that a bit. In C code it has ability to rearrange calls to instrinsics if that doesn't change semantics. In result you get additional optimizations without need to worry about instruction scheduling
3) portability - same C code can be compiler for 32-bit or 64-bit CPU. If you would write asm then you would need to write one for each architecture (or create some pseudo-assembly macros - basically new assembly pseudo-language). This also applies to portability between compilers and OS. ASM syntax can differ between Microsoft Macro assembler for Windows and GNU assembler for Linux/OSX.
Not sure what you mean by Casey checking which compile is running for choosing if they are available. What do you mean by that? Currently he is using MSVC to compile. And MSVC for Intel architecture supports only SSE or AVX intrinsics. He said he will require user to have SSE2 capable CPU, so he freely uses any SSE or SSE2 intrinsic.