I will briefly describe an emulator to explain the main concept mentioned in the question.
Suppose I am using Mame, a video game emulator, and select the old classic arcade "Miss PacMan". Looking at the schematic or looking directly at a PCB inside an arcade video game, it is easy to find the processor : the zilog Z80, the only large chip with 40 pins. Now, if we get the technical data for that processor, we can find the binary encoding for each instruction it can execute. Basically, it get a 8-bit data (value ranging from 0 to 255) which tells the processor what to do. In the case of the emulator, it read the byte (the exact same bytes as would do the Z80 processor inside the original miss pac-man electronic board), determine what a Z80 would do and simulate the instruction.
Some classic video game may have use a x86 processor, similar to the one currently used in most PC. Even when selecting such a game in Mame, the emulator would still read the bytes as found in that game and interpret each one the way the x86 processor would do. In other words, the emulator would not take advantage of the fact that the PC and the emulated game are using a similar processor. It would perform the same steps to emulate any game no matter if the PC on which Mame is running share any similitude with the original game.
You are asking how an interpreter could execute code? The interpreter is a program (the interpreter is just a software, not a physical processor). The wording is effectively confusing. For this sentence to make sense, we would need all the following conditions:
1 - the program to interpret is already in binary, in a machine language that can be executed directly by the processor used in your PC
2 - the program location, the exact address used, is the same as the location that you can reserve in your PC
3 - any library and any I/O occupy the exact same address
When all these condition can be meet, the interpreter could just tell the processor on your PC to stop executing the code from the interpreter but instead, "jump" in the code of the program to be interpreted. Anyone could then say : it is not an interpreter, it is just a launcher.
Maybe such an interpreter which actually does not interpret but let your processor do the real job is still useful in the following way: it could let your processor perform some of the work, but request the generation of an exception when the code to be interpreted is executing some type of instruction. For example, let the code running, but generate a "general protection error" or "trap" or "exception" when trying to execute any of the variant of "IN" or "OUT". The interpreter would take note of the I/O port being written or it would choose a value to give instead of allowing to read a real I/O port. The interpreter would then manage to get the processor "jump" in the program to interpret at the location just after the instruction "IN" or "OUT".
Normally, an interpreter read an ASCII text file, the original source code (which could be Unicode instead of ASCII), determine line by line, word by word, what a compiler would do, then simulate the task on the fly. When the original compiler would need to read many lines to fully understand the current task, the interpreter would also need to read all these lines before being able to simulate the same task.
A big advantage of an interpreter is that it can not crash. Because every instruction is simulated, it is not sensitive to any bug or malicious code. That was a big advantage at the time when computers needed to reboot after encountering any bug, at a time where reboot was taking 10 minutes or more.
Today, with fast SSD to reboot in 5 second and with reliable operating systems which can trap any error in one process and close that process without affecting the stability of the machine, there is less incentive to prefer a slow interpreter over a much faster JIT or much much faster binary executable