Mailchimp how to call mailchimp 3.0 API in javascript
As charliefl noted, this is a CORS issue. MailChimp doesn't support CORS, mostly because it would require you passing your API credentials to the user of the webpage, allowing them to takeover your entire account.
Your two options for MailChimp are to proxy your requests through a server or, for signing people up to your list, you can build a custom signup form that uses a much more restricted API. The caveat of this second method is that it forces all of your subscribes through MailChimp's double opt-in process.