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I'm not a professional, but just an enthusiast but I'll try to simplify the article from my layperson perspective, so take my interpretation with a grain of salt.
The new theory seems to point to how Einstein's theory of gravity considers the "energy-momentum tensor" to be unchanged in all scenarios. The energy-momentum tensor describes the relationship of energy as it changes between various forms, for example a stick of dynamite exploding changes the chemical energy stored in the dynamite into kinetic energy - the force of the explosion - and if you calculated the energy of both they would be equal to the initial chemical energy stored in the unexploded stick of dynamite. Which is called the "law of conservation of energy", that energy cannot be destroyed, only transformed into a different form of energy.
The problem arises in high-energy situations where infinities start to appear in the equations. If you know much about math then infinities can break equations, and often in physics if there are infinities appearing in your equation then it usually means that you're missing something crucial. So scientists can use a technique called renormalization which can apply tweaks to equations to reduce these infinity spikes. At these high-energy situations described, renormalization fails and the equations can't be properly satisfied no matter how you tweak the variables. This is a big problem since a correct theory should be able to come up with answer for all possible situations that might arise within the system it's trying to describe without breaking.
Einstein's field theory - which is a model used to describe spacetime based on the distribution of matter within it - uses the curvature of spacetime, the relationship between stress and energy, and the cosmological constant. The new theory proposed suggests that adding to Einstein's field theory with math that accounts for the relationship between temperature and entropy, and the relationship between charge and interaction changes the equation in such a way that the infinities disappear, even at higher energy levels which traditionally break field theory, and most importantly that it's still consistent with observations.
