Sounds like you accidentally checked the "GIFT THIS ORDER" option on the checkout screen. That's why the game appears as a gift code rather than being added directly to your account.
Just copy the code from the "ORDERS" menu and paste it into the "REDEEM CODE" page. The link to that is here:

https://www.gog.com/redeem

Be careful to not accidentally post it anywhere.

Once you've redeemed the code, the game will be in your account's library. There, click on the "Heroes of Might and Magic III", which will open a menu with the option to download the game's installer (see attached picture).

Oh, either you were given the game as a gift or you accidentally bought it as a gift for yourself. Either way, in that case copy the code you see there, click on "Redeem Code" all the way at the bottom of the website, enter the key and redeem it. Alternatively you can follow the link in the email you got with the gift code, you don't have to manually enter the code this way.

Either way, once the game is redeemed you can see it on your shelf after clicking on "account" and download it from there.

Edit: Ninja'd.
All that blur on your screenshot makes your shelf appear like a porno collection.

I believe you bought the game "as a gift". This means you need to redeem it first, using link and placing the key.
Only then you actually add the game to your account and may proceed to download it as mentioned in previous posts.

And obviously ninjad. twice. I would say No love for non native speakers but both ninjas were foreign nationalities. Damn you EU! :P

Unless I'm mistaken, you may have to redeem the code here first: https://www.gog.com/redeem

Edit: Never mind -- too slow again.

I'll hereby increase the value of this post with virtual dancing.

BOOGIE!

I think that would be less embarassing than openly admitting that I own a copy of Simon the Sorcerer 3D...

Come join this thread for some top ninja action! :P

No, no -- it's about disco dancing now.

NINJA DISCO... OH YEAH!

Hey, whatever happened to your reputation? Have you somehow managed to invoke the wrath of the alternative account scammers?

Most likely, whoever gets their jollies from seeing the arbitrary number go down must have a rather sad existence. This isn't just happening with my rep though as quite a few regulars here are affected as well.

Yeah, I've noticed that regrettable trend... it's just that you never seem to be impolite or provoke anyone so I wondered what might have triggered the rapid decline.

*Sigh*

(At least you're still able to post links...)

You can use the same excuse for both cases, though: "A friend gave it to me as a joke! I never even watched/played it!"

His reputation went down when we caught him with a squid. The ink fetish was a bit much.

posting in the gamegate thread seems to have a similar effect. the "silent majority" at work?

oh shit, i'm offtopic. HI RICKY!

Yes, I got the game as a gift.

I have installed gog downloader now. How do I get my game into there from orders history where I can see my game listed as not yet redeemed. Gift code didn't work when I tried to "reclaim your games. DRM free. I can also see thet my games library currently is empty.