What are CLD and STD for in x86 assembly language? What does DF do?

The direction flag is used to influence the direction in which string instructions offset pointer registers. These are the same instructions that can be used with the REP prefix to repeat the operation.
(Although lods isn’t very useful with rep).

The string instructions are: MOVS (copy mem to mem), STOS (store AL/AX/EAX/RAX), SCAS (scan string), CMPS (compare string), and LODS (load string). There’s also ins/outs for copying between memory and an IO port. Each of these instructions is available in byte, word, dword, and qword operand sizes.

In a nutshell, when the direction flag is 0, the instructions work by incrementing the pointer to the data after every iteration (until ECX is zero or some other condition, depending on the flavour of the REP prefix), while if the flag is 1, the pointer is decremented.

For example, movsd copies a dword from [ds:esi] to [es:edi] (or rdi in 64-bit mode), and does this: (See the “Operation” section in the linked ISA reference manual entry extracted from Intel’s PDFs)

dword [es:edi] = dword [ds:esi]      // 4-byte copy memory to memory
if (DF == 0)
    esi += 4;
    edi += 4;
else  // DF == 1
    esi -= 4;
    edi -= 4;
fi

With a REP prefix, it does this ECX times, and modern x86 CPUs have optimized “fast strings” microcode that does the copying (or stos storing) with 16-byte or 32-byte internal operations. See also this Q&A about memory bandwidth and the ERMSB feature. (Note that only rep stos and rep movs are optimized this way, not repne/repe scas or cmps).