You should see a green button to 'Log in to Steam'. Click this to be redirected to Steam's website where you can log into your Steam account to link this to your Frontierstore.net account.
After the two accounts have been linked, click ‘Claim a Key for Elite Dangerous’ or 'Claim a Key for Elite Dangerous: Horizons Seasons Pass'.
You can now add this key to your Steam account as you would for any other game. Check out this guide for details. You will need to add the key for your Elite Dangerous to Steam before you can add the key for Elite Dangerous: Horizons Season Pass.
Once you've done this, you should be automatically able to install Elite Dangerous via Steam and use the Steam Wallet to make in-game purchases here. If you also add the Horizons Season pass, this should show up as DLC on the Library entry for Elite Dangerous.
Please note: The keys you generate will add your games to any Steam library but WILL NOT GRANT ACCESS TO ELITE: DANGEROUS/HORIZONS without a registered Frontier account and a copy of the game. For this reason, please do not give your Steam key away to a friend! They won’t be able to play without having already bought the game from Steam or from Frontierstore.net.