This is a problem with Google which for some reason identifies normal traffic as automated traffic, similar to that generated by bots.
For all you normal users out there, the problem often occurs when using a VPN or a proxy server. It might even happen when multiple users are sharing the same IP address, (big family, anyone?).
Apparently Google's anti-bot algorithm is quite primitive, and appears to just look at the IP address where the traffic is coming from, and then flat out block all users on a given IP.
It's unfortunate, but detecting bots is close to impossible. However, since most bots are using proxies, one way that some developers deal with them, is to apply a "sledgehammer solution" and block all proxy users and users sharing the same IP. Nevermind that this is actually a great annoyance to users.
Google has a compromise in place, and choose to display a CAPTCHA instead of blocking users completely. Solve this one, and you should be home free for the rest of your session. If you use private browsing, such as incognito, then you may run into these CAPTCHAs more often.
In Google's case however, it should be easy to differentiate between actual bots and probable legit users. One way to do it, is to look at the search phrase, and identify keywords used by spammers. That would include: "email", "forum", "guestbook", and many other website footprints that normal users are almost never using.
CAPTCHAs are difficult for humans to solve, and often takes 1-3 minutes to pass, (even more if you fail a few times). The ones used by Google, where you have to identify objects in photos are particularly bad, because the pictures are often very poor quality, making it hard to see what is actually in the photo.
Avoiding Google's CAPTCHAs
As it typically is with Google's official help files, they are not very helpful, and does not provide a way to actually solve the problem for all the users who are wrongly being affected by this. Rather they suggest that users disable their VPN, which would defeat the users security.
The blocking might be related to your use of a VPN browser plugin or program. You might try uninstalling the VPN from your computer or network and see if that makes a difference.
Instead of compromising your security, you can try to limit your use of incognito (private browsing) windows, and instead keep your window open. After solving a CAPTCHA once, you should generally not be presented with more of them in your session.
Also, avoid using exotic search commands such as "intitle:" or "site:" as they might trigger a CAPTCHA.
Signing in to your Google account can solve the problem. Letting Google know who you are is a small security risk, and likely not something everyone is willing to do.
You can also use duck duck go instead of Google, but the results may be worse.