Mfcttrf

When and why did hexadecimal representation become more common than octal for displaying and printing out multi-bit binary fields?

Addressing the "why" part of the question - from my point of view as an assembly-code programmer on PDP-11 and VAX, the "standard" radix is most usefully chosen to match the instruction layout.

PDP-11 had 8 registers and 8 operand-mode indicators. Its double-operand instruction layout was

1 bit generally byte/word indicator (b)
3 bits opcode (o)
3 bits source mode (s)
3 bits source register (r)
3 bits destination mode (d)
3 bits destination register (R)

making octal the perfect way to express it:

booosssrrrdddRRR

The VAX, on the other hand, had 16 registers and 16 bits for operand mode (though some combinations were used for short literals). A basic operand specifier, in the variable-length instruction format was

4 bits mode (m)
4 bits register (r)

thus hex was perfect to express these.

mmmmrrrr

Of course, the larger address space used on VAX gives other advantages to hex: fewer characters in an address. This might have some bearing on "when".

Minicomputers and mainframes typically used octal, as many early mainframes had word sizes that were a multiple of 3 bits, and so did some minis. Operators and engineers within those environments became used to this, so even power-of-two word size minicomputers kept using octal.

Microcomputers, however, almost always had power-of-two word sizes for both address and data buses (or at least, a multiple of four bits), and there was a whole new generation of users who were not mentally locked into the mainframe/mini way of thinking. It was thus natural to start using hexadecimal instead.

You'll probably find, therefore, that hexadecimal rose to prominence about when microcomputers did, in the mid to late 1970s.

When and why

That's quite close tied to the IBM /360 and its introduction in 1964. The /360 is based on the use of an 8 bit byte, 32 bit word (16 bit half word) and 24 bit address. Thus all basic memory items were multiples of 8 bit units - which are, without any remainder, best be displayed in Hex.

Before that size of bytes, half words and words were (more often than not) multiples of 3, which works quite fine with octal. After all, grown up with decimal it's way less mental work to not use two numbers, than to learn six more. It seams more natural, doesn't it?

After that next to all new designs switched to 8 bit bytes to allow easy data exchange with IBM mainframes. This happened even faster for mini computers as they where usually supplementary systems to (/360ish) mainframes.

See also this question about the rational of 36 bit designs. While not a true duplicate, it's quite related here.