RGB and CMYK are very different color models. For instance, RGB is additive: you basically start with black and add colors. You combine all colors to make white. But CMYK is subtractive: you start with white and add colors to generate black. If you think about it, it makes sense: Monitors are black by default, when there is no light, the screen is black. Paper is generally white by default, and colors are added to it with printing.
Understanding this, it is then easy to understand that monitors are RGB, and printers are CMYK. You can't change a monitor to CMYK, since it doesn't work that way, literally.
Calibration simply sets your monitor to a neutral state, so that it matches a reference. This allows you have your monitor in a known state, and ensures consistency.
Photoshop shows things on screen in RGB (usually Adobe RGB) for obvious reasons, but when you are ready to send to pre-press, you convert to CMYK, which is what the vendor's printer is expecting. This conversion needs a 'translation' that tells it how to match the RGB colors to CMYK colors. Photoshop has a table, but it needs to know how that table compares to the printer.This is called a color profile or ICC profile.
The ICC profile allows Photoshop to interpret colors on the screen to colors on the printer, and also you to interpret what you see on screen to be close to or indicative of what is printed. The thing is, these profiles are very specific, down to the printer model, ink used and paper used. For home printers, there are generic profiles that are usually better than nothing, but for pre-press, you need to get sample files and prints, to compare on your monitor so you can dial in your calibration to match the output of the target printer as closely as you can. Note that this is needed for each vendor/printer, and some even do it before each 'run', especially if a run is going to be extensive ($$).
So, your designer is partly right: he can not rely just on the calibration of your monitor, since that is calibrated to a known standard for monitors. You need to go the next step and get the ICC profile and then compare the output to what the printer outputs. Many calibration hardware can do this as well, known as printer calibration, where an images is printed, then scanned by the colorimeter, and an ICC profile is created.
No, you don't calibrate your monitor to CMYK vs RGB. You first need to calibrate you monitor using a hardware calibrator (not the squint method) , then get a ICC profile from the printer, and finally a sample file and print combination from the print shop to compare.