How do I get card drops in the Steam summer sale game?

All you have to do to get the free cards from the game is play the game, and the game plays itself even if you're not actively watching/playing it. Even if you load the game once and then close the tab, you'll still get cards from "playing." (I've gotten one card so far this way, after loading the game once and having the tab just lock up on me)

The maximum number of free cards from the game is 3 per day, and they're given to you at random intervals over the course of the 24 hour period.

The reason to really "play" the game is to help unlock the community goals and allow for more games to go on sale.

It says on the page so here is what it says:

Craft a game badge

Starting June 1st, crafting a game badge earned you a Summer Sale trading card rather than a coupon, so if you crafted a badge in the last week you’re already ahead of the game. During the sale, each game badge you craft will get you an additional Summer Sale trading card.

Make a purchase

As an extra reward for purchasing, for every $9 USD spent during the sale, you’ll receive a Summer Sale trading card. From your Summer Sale badge page select “How do I get more cards?” to track your purchasing progress.

Play the Monster Summer Game

If you’re Steam Level 5 and above, you’ll get a Summer Sale trading card at random intervals while playing, up to three per 24-hour period. (If you reach Steam Level 5 before the end of the sale, you’ll be granted the cards you earned by playing the Monster Summer Game).

Trade for them

That’s why they’re called trading cards! Find friends with other cards and make trades. Your Monster Card badge page shows which friends have the cards you need.

Buy them from the Community Market

Sell the cards you don’t need and buy the ones you do, all with other Community members. Turn those duplicate cards into Steam Wallet funds to buy games.