It's false - if you want to process many bits at once, you use SIMD instructions, which you have on 32-bit too.
Generally, other things equal, going from 32 to 64 bit word size makes things slower because pointers suddenly double in size, and you can fit fewer program objects in your caches.
In practice there are a slew of miscellaneous improvements in the new 64-bit revision of an instruction set that makes up for some of the slowdown. This is less pronounced on ARM, but was big on x86 because a big flaw, the register count, was fixed.
However, ARMv8 also increased the power of the SIMD portion of the chip, going from a 64 bit wide set of registers to one that's 128 bits wide. So SIMD-heavy code will see a decent speedup because you can process more data per clock.
Generally, other things equal, going from 32 to 64 bit word size makes things slower because pointers suddenly double in size, and you can fit fewer program objects in your caches.
In practice there are a slew of miscellaneous improvements in the new 64-bit revision of an instruction set that makes up for some of the slowdown. This is less pronounced on ARM, but was big on x86 because a big flaw, the register count, was fixed.