I just spent 45 minutes struggling to redeem a physical Origin gift card for a Sims expansion pack for my daughter - the Origin, Sims, and EA sites were of zero help. I've been Googling things like "how do i use an Origin Gift Card" to no avail, but I stumbled across the solution myself that may work for you...
So - as origin says, you have to add the gift card credit to you eWallet...(for some unknown reason you can't just use the gift card directly as a payment method when you check out like you can with 90% of online retailers)
First and foremost. Ignore the instructions on the back of the card that say only to "go to thesims.com/redeem and follow instructions" (this will get you nowhere...nothing makes sense from there - it asks for a 'product code' which is not helpful)
Instead, what you want to do is get to your ea.com account page. I got there randomly by going to help.ea.com, there may be a faster way to get there. I clicked on Origin --> PC --> then you want to click on one of the rectangles called 'manage my account' and then 'payment method' - which should put you on a page inside myaccount.ea.com with a big gray menu of account management options (About Me, Security, Privacy Settings, etc etc) - click on the one called PAYMENT METHODS, and from here you should see an option to add to your EA Wallet.
They'll ask for the 'Card Code' - of course there are multiple numbers on the gift card, none of them with that label (thanks EA!) - you'll want to enter in the scratch off PIN, then the funds should appear in the EAWallet, and should be usable at checkout.