reCAPTCHA is a pop-up window. If you have pop-ups disabled in your browser, this feature will not work properly.
If your internet connection is slow, it may still be trying to load more images while you are trying to move on to the next step.
If you are repeatedly using the same payment page in the same browser, the browser recognizes this and will add extra layers to the verification process to ensure you are not a robot.
Make sure your browser is up to date.
Try another browser. IE is the most temperamental browser with the most reported incidents whereas Chrome seems to be the most reliable.
If you are using Chrome, being signed into your Google Account may boost confidence that you are a real person.
If reCAPTCHA seems to be stuck and won't let you move forward, try clearing your cache and cookies.