I think backward compatibility was the main aim. Intel tried redesigning the architecture as 64-bit native (Itanium), but AMD done a better job at backward compatibility - and intel eventually adopted it as x86-64.

The amd64 design could run most 32-bit code with minimal changes. All the 32-bit instructions had the same encoding, besides push/pop which instead acted on 64-bits. The 64-bit instructions were basically opt-in, though a few opcodes (0x40..0x4F) had to be deprecated for the REX prefix.