I think Ed's comment about the software vs. hardware view of the process is on target. The sentence in question causes some confusion by not identifying that the second and third bytes must be held in internal registers / latches that are not visible to the programmer. It is from these internal registers / latches that the data is routed as described in the referenced text. Given the two phase nature of the 6502 machine cycle, it is possible to reduce the number of temporary registers to reduce the transistor count. Since latches are the primary memory devices used within the 6502, it is possible to transfer one operand, from a single internal temporary operand latch, to the low address latch, and simultaneously transfer the second byte of the address from a data input buffer, implemented as a latch, to the high address latch. Simultaneously, this 16-bit address can be loaded into the program counter. Refer to the 6502 block diagram provided at this
link.
This flow through feature of latches can be used very effectively to reduce the number of memory elements required. However, it requires precise control of the latch control signals, which can be difficult to achieve when high rates, or short delays, are required. Thus, modern design practices avoid the use of latches and instead rely on registers, which do not allow the flow through action described above.
In the case of my 6502/65C02 FPGA-based emulation, I use two registers which I name OP1 and OP2, which hold the first and second operand bytes of an instruction, respectively. When I load OP1, my implementation either zeros OP2, or loads it with all 1s based on the most significant bit of the operand byte being loaded into OP1, i.e., sign extension. OP2 is the sign-extension of the OP1 for all branch instructions, and simplifies the implementation of the PC-relative addressing associated with the 6502/65C02 branch instructions.
When I "execute" the instruction after all required operands are present in the operand registers, my implementation effectively performs the actions described in the referenced text: OP2, OP1 are either used to modify the PC, generate a direct / indirect operand address with or without indexing, or supply an immediate data to the ALU which may simply pass it through to the A, X, Y, or P registers.