This project, like many areas in technology (especially new or cutting-edge), are not well addressed in laws, regulations, or legal frameworks. The courts of the world have not provided much clarity either, but certainly have not provided any clarity on this particular project.
You did not reference your view in relation to a particular country or legal authority, but the site is hosted in the U.S. so that is the most relevant here.
The content is not hosted by hostyoself.com, it's hosted by the systems owned by the content sharer. The legal landscape has certainly shifted providing less coverage for user-generated (or uploaded) content but there are still some protections in place for solutions that do in fact host user content. But again, that's not the case here.
hostyoself.com is simply forwarding requests to the relevant destination. Not unlikely any other traffic routing or forwarding technology. In fact, the proxy comparison is particularly apt, and a solid defense.
Of course none of this absolves hostyoself.com from complete inaction. I could certainly see them compelled to cease forwarding requests to some destinations. But I do not see a clear path for hostyoself.com to be liable for the content a particular user is hosting. The user in question is the one in possession and performing the hosting/share of the content.
Of course... the courts are the ones who would decide in such a case.
That's not to say they couldn't use the same approach to legally questionably or illegal content: If they have a proper complaint or takedown request, they stop forwarding to that content host.
The drag and drop interface isn't to upload the content to be hosted like a traditional server (what's the point then?) It looks like (without digging through code) it's so they can parse the file-structure and generate the URIs to forward requests.
I haven't looked at the code either. But the hosting seems to be handled in JavaScript. There are no required clients (though there does seem to be an optional one); this is all completed in the browser.
So you're not "uploading" content, you're providing the content to the client-side application so it can be shared through that same client-side solution.
you <-> hostyoself <-> someone else
where all data goes through hostyoself. This is analogous to ngrok, but at file-system level rather than TCP.