We have to go by what people say and show us, but it seems like you're really stretching the limits of your knowledge to do this safely. That said, you have to start somewhere, so here goes.
First, the "old switch" is a 3-way switch. You surely know this light is controlled from 2+ locations. I suggest boning up on how these circuits work, so that you know what you're looking at and so you'll be able to reassign these wires to what the smart-switches need, since I have a feeling the other end will be involved too.
Traditional 3-ways have 2 travelers and 1 common. The 2 travelers are always in the same cable, so it's that red and the black that is with it. Clearly the other black is in a different cable since cables don't have 2 blacks.
They all have whites though, where are they? No doubt pushed into the back of the box and nutted together, these are your neutrals - probably. You'll need to pigtail off those to obtain your smart switch's neutral.
Traditional switches don't have neutral screws because they don't use neutral.
Note that wire color does not correspond to wire function. 3-way circuits especially. For instance your travelers are red-black; I prefer them yellow-yellow and will typically mark them that way with colored tape.
Hopefully armed with that knowledge you can fit what you have to the wiring diagrams in the instructions. You are required to obey the instructions when installing the product (that's in Code; NEC 110.3).
It may be worthwhile to go to the local library and get a book on wiring home electrical, and read a great bulk of it through. This will give you a well-rounded primer on the subject. Won't answer every question, but we're here.