Yes, remove your SIM card before selling. If you have activated the new phone already, skip this bit, but you could have removed the old SIM card and put it in the new phone so long as they were the same size and both 4G.
Blo93, congratulations on your new phone. As the SIM card itself, does not contain personal information, you do not need to fear leaving the card in the old phone. You will want to be certain that you do a master reset on the old phone to wipe it back to factory settings so that all of your personal information is erased from the phone. You can do this by going to the Settings, General, Reset and selecting the reset all selecting the Reset all Content and Settings.
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I think it can be sold with the phone simply because they already activated the other phone, thus rendering that SIM card null. I still wouldn't sell a phone with an old SIM card. That to me is up there with selling an old computer with the hard drive.