Encrypted searches are (more) secure from 3rd party snoopers. Web sites you visit, but also intermediaries like internet cafés and ISPs can see and read the traffic to and from your computer. If that traffic is encrypted, that eavesdropping capability won't do them much good. A big issue with this (although there are several) is the fact that web sites you visit collect information from your computer and web browser that allows them to recognize you on repeat visits and check on your previous activity to tailor what they're going to show you on their web pages you navigate to. They can do the latter, because the content of any web site can be (and often is) determined on the fly.
This 'tailoring' can be helpful to you as user, because web sites can serve you info of known interest to you, and omit that which you haven't shown to be interested in on previous visits. But it also affects your privacy, because the information that your web browsing 'leaks' all over the internet can and will be aggregated in comprehensive personal dossiers, which are increasingly traded among interested parties in the industry. Combine that with credit reports etc., and you'd be surprised what's known about you. Now imagine that these repositories are hacked by malfeasants, assuming for a moment that they cannot buy them outright (a questionable proposition to say the least), and you could be forgiven for thinking you might as well throw the keys to your life out on the street...
To return to Google's search encryption, that was started in part to thwart internet criminals from stealing your info and using that to get even more on you by making it easier to track you on the Internet in ways similar to what 'regular' web sites do. Alternatively, they're after info that can be used to fool you into releasing it yourself (phishing), like passwords etc. that would allow them to pilfer or otherwise mess with your accounts. Encrypting search terms and search results is by no means full proof, but it's a big step in improving user privacy and security.
