I am trying to understand how floating point conversion is handled at the low level. So based on my understanding, this is implemented in hardware. So, for example, SSE provides the instruction cvttss2si which converts a float to an int.
But my question is: was the floating point conversion always handled this way? What about before the invention of FPU and SSE, was the calculation done manually using Assembly code?
It depends on the processor, and there have been a huge number of different processors over the years.
FPU stands for "floating-point unit". It's a more or less generic term that can refer to a floating-point hardware unit for any computer system. Some systems might have floating-point operations built into the CPU. Others might have a separate chip. Yet others might not have hardware floating-point support at all. If you specify a floating-point conversion in your code, the compiler will generate whatever CPU instructions are needed to perform the necessary computation. On some systems, that might be a call to a subroutine that does whatever bit manipulations are needed.
SSE stands for "Streaming SIMD Extensions", and is specific to the x86 family of CPUs. For non-x86 CPUs, there's no "before" or "after" SSE; SSE simply doesn't apply.