The form number is 4457 not 4473. You will need to take your rifle(s) you intend to take with you to a CBP office and fill out the 4457 there and have it signed by a CBP officer prior to your trip. And depending on the airline, you may need to do this several weeks in advance.
Each airline has it's own process and even each airport seems to do things a little differently. But if you're going on Delta, it's merely show up with your rifles and yes the 4457s at the airport. They will want you to put a red card in the rifle case that you're confirming the gun(s) are unloaded.
From there one way or another, the rifles will need to go to TSA. And you will go there with them. Again each officer in each airport seems to have their own way of doing things. Some can be very edgy about this while others have been there and done that and are calm. But in the end they want to make sure you don't have an explosive and rifles are unloaded. I personally remove the bolts from the rifles and have them in there own spot in the case.
I also put my ammunition in a different and obviously locked case.
So long as all is well, this should be the last you deal with your rifles until you get to Africa. But be prepared. My last trip, TSA called on us at the gate to confirm we were traveling with rifles. Yah, that seemed stupid to me too since we had already met with their colleagues. But have your 4457's handy, stay calm and be polite and all will be fine.
Now having said all of that, again I was presuming you will be on Delta. But it's been awhile since I've flown on Delta, so this is certainly subject to correction. And, things change a little in the process with other airlines.
I recommend once you're ready to book your flights you check back here on AH to get the lowdown on the airline and itinerary.
