Free storage that they allocate but that you don't actually use costs epsilon more than nothing at all at the margin. The part you do actually use costs them about 10 cents a month on Amazon's published pricing, and one could reasonably assume they have a better deal than publicly available. At a buck a year and what I make out to be a viral factor of something like 20, you're approximately the cheapest cost of customer acquisition ever. (If that worked for my business at scale I would be rolling in it, and my LTV is much lower than Dropbox's is.)
How they make money is fairly simple: use you as a customer acquisition channel to get people who have money using the service. Charge those people many, many times what servicing them actually costs. Does it matter if there are cheaper options? Pfft, no. There are cheaper caffeine delivery vehicles than Starbucks coffee, but since people don't choose coffee primarily based on price, if they hook one more weekday latte drinker that gets them a LTV of several thousand bucks at a margin of ZOINKS%. Dropbox isn't Starbucks, but their unit economics can be reasonably assumed to be "quite favorable indeed."
How they make money is fairly simple: use you as a customer acquisition channel to get people who have money using the service. Charge those people many, many times what servicing them actually costs. Does it matter if there are cheaper options? Pfft, no. There are cheaper caffeine delivery vehicles than Starbucks coffee, but since people don't choose coffee primarily based on price, if they hook one more weekday latte drinker that gets them a LTV of several thousand bucks at a margin of ZOINKS%. Dropbox isn't Starbucks, but their unit economics can be reasonably assumed to be "quite favorable indeed."