These crawls are part of an effort to archive pages as they are created and archive the pages that they refer to. That way, as the pages that are referenced are changed or taken from the web, a link to the version that was live when the page was written will be preserved.
Then the Internet Archive hopes that references to these archived pages will be put in place of a link that would be otherwise be broken, or a companion link to allow people to see what was originally intended by a page's authors.
After you protect your Tweets, only you and your followers can read your updates or see your Tweets in Twitter search.
If you at one time had public Tweets (before protecting your Tweets), those Tweets will no longer be public or appear in public Twitter search results. However, unprotecting your Tweets will cause any previously protected Tweets to be made public.
To find out how to remove your updates from Google search, check out this article. Twitter does not have the ability to remove content on websites other than twitter.com.
What If my Tweets have never been public?
If your Tweets have never been public (in other words, you've always had your Tweets protected in your settings), your updates should never show up in Twitter search or other public search engines.
