The word "unsupported" implies that the argument being made lacks the necessary information to back it up. Therefore, "unsupported assertions" are claims that are made during a conversation or a debate, but which are made without enough evidence to back them up.
Making unsupported assertions during a debate can lead to trouble for the participants. When such a claim is made, it is stated as a fact. However, for the other team to accept it as such, it is necessary to show that there is enough evidence to support it. Being unable to provide this evidence not only renders the argument useless, but can damage the opinion that the audience has of the team that put forward the claim.
