Trying to book a hotel using booking.com it states that "Additional charges may apply"
What are those charges?
How to calculate them?
Do I have to Pay them in cash?
It really depends and even Booking.com does not know which is why they use the word may. Generally speaking there are additional fees that they either cannot collect or cannot calculate due to complex rules that need more information than provided.
Sometimes there are no additional fees at all but often there is something charged by the locality. From the one example you show, Booking.com already calculated with local city tax which is included and known at the time. The last time I booked with them, the additional fee was 0.5 Eur per night which was a Tourist Tax with it being applicable to me as a tourist but probably not be applicable to someone local or on a different visa. Some hotels will collect them at check-in but most at checkout when you settle the bill which makes it easy to pay by card. I did not encounter a time when it was required to pay cash but that does not mean it is impossible.
To calculate them you can do some research about your destination but the more certain way to know is to just call the hotel in question.
As booking.com states, requirements made directly to the hotel, like extra beds, cots for babies or pets allowed at extra pay, will be charged by the hotel.
Those are charges you should know about as you requested them or agreed to them.
Tourist tax is also to be paid directly to the hotel but is often already mentioned on the booking page.
An extra charge not yet know may be cleaning cost after illegal smoking or having pets in the room when not allowed.
How you pay these charges depends on the hotel, often they give you the options cash or cards but some only take one kind of payment, booking.com mostly mentions that in the hotel information on the booking page.
Cleaning charges can be taken through the credit card with which you secured the booking.