I had to google ELI5 :-)

First:

• Programs are sequences of commands, conditions and different forms of repetitions. The form of these programs follows a very strict language.
• Types are a kind of language describing the words and numbers used in programming languages (or in math).
• Interpreters are programs that look at the description of programs and do what the commands say.

OK, now what the result means is that Brown and Palsberg achieve to create a system of types that allows to formally describe the objects (words, numbers) used in interpreters that can interpret themselves.

Why is that astonishing? Interpreters have been around for a long time, or? Including those that interpret themselves (which is called bootstrapping). Yes, but most of these use only very simple types. The power of the types used in a programming languages limits the way one can reason about the programs in the language. For the simple lamguages the interpreted programs are just a bunch of words. It is not possible to state that the bunch of words is a valid program. The System F mentioned above is quite powerful mathematically speaking. Fω even more. But both were not able to describe a program which interprets itself as valid program. This was achieved by System U.