One of the main reasons is that you can see the site's funds. It's all public. Most of the sites share their addresses, even if they don't, you can basically find it by just checking a few addresses on the blockchain. This is especially important when checking out, for example, an exchange site. If the exchange does not have even 1 BTC but is advertising themselves as the biggest and the best exchange, that should raise some flags. So, long short story. Transparency, whether they like it or not is probably my main reason.
So you want like, to be able to see the amount of bitcoins they have in their wallet, and for example statistics on how the site is doing?
Yes. The example I gave was for an exchange. The situation depends on what kind of platform you'd like to register to. If we want to go by your example, a gambling site. You'd like to see that public statistics, their hash system (provably fair), their userbase and if possible, their current bank (their main BTC address) which is going to handle the payments.
