Curb check-in is the same as curbside check-in. You check your bags to your final destination outside the airport, and don't have to schlep your bags through the check-in line inside.
There's no more sentence actually. The sentence about which I asked is the title for a video discription, and I don't think the video is focusing on the curbside check-in, it looks like explaining checking-in in general.
It has an implied "from" in front of it - i.e. you should read that as "from curb to check-in", meaning "this section of the video will deal with the portion of your journey beginning after you leave your vehicle and proceeding until you check in at the airline".