I don't think this question is worthless to this site. While it has a misleading title ("Why use dropdowns over textboxes in this case" would be better) I had thought it worth answering. I ask that you consider reopening it.
I think the best case I can make is to simply show the answer I would have given:
This is for mouse driven people. Keyboard driven people can still type.
Back in the day, when Apple was first selling computers that came with mice, Microsoft sold Windows to people who didn't have mice, yet. Because of that MS designed Windows so that it could work with and without a mouse. They continue this tradition of supporting both even today.
That is the mentality behind what you see here. Yes, a text field could stand in here. Yes, it could also reject invalid input without an error message. But by doing it this way it's supporting both keyboard and mouse driven people. Even if that isn't all that obvious to the keyboard people.
If you're wondering where to find this date calculator, and you're keyboard driven like me:
Press ⊞ Win+R then type: calc Enter. Once the calculator launches press alt+H and press ▼ until Date Calculation is highlighted, then press Enter.
If Difference between days is highlighted press ▼ until Add or subtract days is highlighted. Then press Tab until the Days dropdown is highlighted.
At that point you can type in a number without getting distracted by a bunch of other numbers. All without touching the mouse. It's almost, but not quiet, like using a textbox. No blinking cursor, no Backspace. But you can type.
You can also do all this with only 8 clicks of the mouse, which keeps the mouse driven people happy. A textbox would force them to touch the keyboard.