Why does my site url appear in Google Webmaster Tools as remove url request? The Removal Type is "SafeSearch page filtering", but I did not make the request. What is the cause of this?
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Based on the Removal Type column in your screen shot being "SafeSearch page filtering", and the fact that acknowledge that it's adult content, I think that likely somebody has reported your site to Google as being adult content, or offensive in some manner. Users can submit adult content URLs at https://www.google.com/webmasters/tools/safesearch.
Google has a "SafeSearch" mode that people can turn on either through their Google Search settings, or through their browser settings, so that users can filter out objectionable content, for themselves and children, etc Read more here.
Now, I've never actually seen this type of removal request personally (because I've never dealt with content of such a nature), but I'd bet that your site will still be indexed normally when Safe Search is turned off, but likely won't appear for users who are filtering explicit or adult content, etc.
So to answer your question, the Removal Request is coming from other people, and there's likely nothing you can do about it, except remove any adult or explicit content. In other words, it's a feature not a bug.