Pack it anywhere you like. I've done it at home, I've had a shop pack it, and I've packed it (several times) right there in the airport.
And here's how the process goes when you get to the airport with your bicycle and luggage:
1) Ensure that your bicycle is safely packed in the box (whether you packed it at home or there at the airport); ensure that you've got only one other piece of checked luggage; ensure that there is nothing they might consider dangerous in your carry on luggage.
2) Place all your pieces of luggage on a trolley for easier transportation around the airport.
3) Walk, with your trolley full of luggage to the airline which you are flying with to check in and get your boarding pass.
4) Your bicycle on the trolley will pose a bit of a problem as you round the corners of the line to get your boarding pass, but if I can do it, you can do it. Just smile a lot and apologize if you bump into anyone or knock things over.
5) When you get to the desk they will ask you if you packed it all yourself and if you left it unattended anywhere. Hopefully you did pack it yourself, and didn't leave it alone somewhere. They will give you a sticker to stick to the box and to attach to your other piece of checked luggage. You should take this moment to fill out address tags which you can tape to the box and so on. They will also give you your boarding pass complete with little tags for claiming your luggage later if necessary. They will point them out to you and tell you about them. Don't lose those! If your luggage doesn't show up at the other end, they are the proof that you had luggage and aren't trying to swindle the airline.
6) What happens next will depend on the airport. You may be directed to take everything to the X-Ray machine which will be within sight. You will have to wait in another line, but as you get closer someone will likely come along and take you (and your bicycle and everything) to another side of the machine to send it through there along with all the other oversized luggage. Or after you finish getting your boarding pass the person you are dealing with may signal someone to come over and that person will escort you (and your bicycle and everything) to another part of the building, where you will wave good-bye to your bicycle as they take it away. Or you will be sent through customs with it (this happens to me when I fly from Canada to the US) where they will ask you 1001 questions about the bicycle and cycling in general, and then you will be escorted by a security guard or someone to where you need to drop it off.
I'll just mention here, that when I fly without my bicycle I am a nameless face in the crowd and I get treated like all the rest of the nameless faces in the crowd. But when I fly with my bicycle all of a sudden I become somehow "special" and I get escorted all over the airport, and put to the front of lines and everything. It's very strange ... but nice!! I'm definitely not complaining!!