As explained in these forum-based instructions, once on the store page you’ll need to log in, go to your account dashboard and find the ‘Partner Keys’ option. Click on that, then find the ‘Claim an Elite: Dangerous key’ option.
Doing this will generate a key for Steam, which you can redeem on there in the usual manner.
The forum post adds an important note for anybody who might have been thinking “ooh, an extra key, I can give that to someone else”: Note: the key you generate will add Elite: Dangerous to any Steam library but WILL NOT grant access to the game without an existing registered Frontier account and copy of the game. For this reason, please don’t give your keys away to friends!
Later in the thread, it’s confirmed that any Elite: Dangerous accounts linked to Steam will subsequently be giving a cut (30% is the standard) of any future expansion or cosmetic item purchases to Valve. Something to be aware of if you’d prefer Frontier to keep getting the full amount from anything you might buy later on.