Based upon what I have seen browsing the website, the short answer is that Rabbit is not allowed to do it. I suspect that they are trying to get away with it for the same reasons YouTube gets away with hosting infringing content: Internet service providers, such as webhosts, are protected by D.M.C.A. safe harbor provisions codified into U.S.C. Title 17 §512.
These provisions exist so that webhosts and other such internet service providers are not held liable for the infringing actions of end users without their knowledge. The internet service provider is required to remove the infringing content upon discovery, and document a procedure for removing any infringing content at the copyright holder's request.
The reason I suspect this is based on Rabbit's terms of service which has specific provisions against using the service for the hosting of infringing content, under section II. User Content, subsection A. Non-infringing Content Sharing, which includes a D.M.C.A. takedown notification procedure. Here are some especially noteworthy excerpts:
The Company Service offers Users the opportunity to share content with each other. Company encourages such sharing but prohibits copyright infringement or the infringement of other intellectual property rights through the Company Service.
Content Removal Policy.
We will respond to notices of alleged copyright infringement that comply with applicable law and are properly provided to us. If a rights holder believes that User Content has been copied in a way that constitutes copyright infringement, such rights holder or its agent or designee should provide our copyright agent with the following information in accordance with the Digital Millennium Copyright Act:
Notice that under How Does Rabbit Work section of the F.A.Q. they claim:
Rabbit allows you to share content with friends, video and audio chat, and text chat within a Rabbit room. You can share content with a virtual browser (Rabbitcast) or by sharing a Chrome tab using our Share on Rabbit extension.
Basically, they seem to be claiming that since the users are sharing the content, it is not their fault if their services are misused.
That much would probably be fine, but unfortunately for Rabbit, they are not in perfect compliance with the law because they are advertising their service for the express purpose of sharing television shows and movies for which an end user is unlikely to own distribution rights, so it could be construed that it is at their direction that the infringement is taking place. If such is the case, then they are not eligible for U.S.C. Title 17 §512 protection. Advertising services in such a way is what got Grokester in trouble in M.G.M. Studios, Inc v. Grokester, Ltd. I would not count on this service lasting, or at least not how it is now, since they seem to be liable for contributory infringement.