The batteries we use on our drones are LiPo batteries which need proper care. When you go on a plane the pressure changes, and it is stressful for the battery. If the battery is at storage voltage it helps because the charge inside can fluctuate freely, but when it's fully charged it might become unstable and dangerous.
LiIon batteries you use on your phones and tablets aren't the same, they can't provide such huge currents, instead, they provide more capacity. These batteries aren't that dangerous to transport while charged.
This is a myth perpetuated in drone forums.
As I said it's not suggested by any airline or any air safety authority.
You can do it, but there's no reason that you should.
And lots of laptops (and possibly other devices) use LiPo batteries too.
While airlines would not check for this, it is still good practice to do it, even in the manual of the drone it says to "Discharge the battery to 30% for transportation, this can be done by flying the aircraft outdoors" or something similar.
DJI manuals have contributed to a number of myths, including this one.
Here's a little background to this one.
According to DJI's Intelligent Flight Battery Safety Guidelines ...
Travel Notice
1. Before carrying the batteries on an airline flight, they must first be discharged to a battery level lower than 30%
Extensive searching doesn't show any airline in the world asking for this.
The only place I find any mention of it is in the IATA's guidance document for shipping lithium batteries as air cargo:
Documents and downloads complementing the current edition of the IATA DGR.
www.iata.org
So if you are shipping a pallet load as air cargo, then you have to discharge.
The International Air Transport Association (IATA) is the international body that supports aviation with global standards for airline safety, security, efficiency and sustainability.
All airline lithium battery policies are based on the IATA standards.
And the IATA's guidelines for Passengers Traveling With Lithium Batteries:
Documents and downloads complementing the current edition of the IATA DGR.
www.iata.org
.... makes absolutely no mention of discharging batteries before traveling by plane.