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I have been looking at some Verilog testbench code that heavily uses $readmemh and $writememh .

I have a vague understanding that these functions basically read to and write from memory. What is their specific function and how do they work?

I agree its not too easy to find something about readmem/writemem. You can find a little bit here: https://www.fullchipdesign.com/readmemh.htm

Anyway there is not too much to say about these functions, the syntax is:

$readmem[hb]("File", ArrayName, StartAddr, EndAddr)
$writemem[hb]("File", ArrayName, StartAddr, EndAddr)
Verilog is very picky about the file format, the number of bit in the text file have to match the number of bits in the array.
I recommend you play around a little bit by defining an array, filling it up with data write it out with writememh/writememb and print it out afterwards.
Something like this should get you started (not tried out!).
integer i;
reg [7:0] memory [0:15]; // 8 bit memory with 16 entries
initial begin
    for (i=0; i<16; i++) begin
        memory[i] = i;
    $writememb("memory_binary.txt", memory);
    $writememh("memory_hex.txt", memory);
                As well, the tools can be picky about finding the path.  Project F: FPGA Dev has some information on this.  The 'fool proof' way is to provide a full path.  But full paths are foolish.  I believe that "File" can be supplied as a module parameter and often this might be a better mechanics.  See: Passing string as a parameter.
– artless noise
                Jan 3 at 18:43
                Another reference to the file format. 1364-2005 $readmemh where order in the file is important.   Altera/Intel provide JTAG mechanics to update the table at run time ("In-System Modifiable Memories and Constants"). Eg. Dual port ram
– artless noise
                Jan 3 at 19:39