This submission's headline is wildly overstated. The paper seems at first reading anyway to be a quite reasonable piece of work suggesting a mechanism by which a particular defensible peptide works. The authors themselves don't talk about cancer cures as well they shouldn't.

For one thing this peptide was studied in vitro - against cells in a dish. Many many things work against cancer cells in vitro. It's only active at micromolar levels which is really not very potent compared to most drugs. And when you compare its activity against cultured tumor cell lines versus noncancerous cells the window is about five-fold which is likely to be real but also not very wide for therapeutic use.
The next step in testing a potential anticancer compound (besides running it against a much greater variety of cell lines) is to see how it behaves in a real living system. You'd want to see what the pharmacokinetics are like - how long does it last in the bloodstream when given to a mouse for example by different routes? What concentrations does it reach in various tissues? While you're doing that, you'll also get a first look at safety, since at the higher PK doses you'll be watching for side effects.