Relativity = gravity is represented by the curvature of spacetime. Curvature is linear, R. The formula treats curvature linearly. As things get closer and curvature spikes, the math just scales at a 1:1 rate
Quadratic gravity = Squares the curvature. Doesn't really change things much when everything is far apart, but heavily changes things when everything is close together.
Pros: prevents infinities and other problems when trying to reconcile quantum theory with relativity ("makes the theory renormalizable"). E.g. you don't want to calculate "if I add up the probabilities of all of these possible routes to some specific event, what are the odds that it happens?" -> "Infinity percent odds". That's... a problem. Renormalization is a trick for electromagnetism that prevents this by letting the infinities cancel out. But it doesn't work with linear curvature - gravitons carry energy, which creates gravity, which carries more energy... it explodes, and renormalization attempts just create new infinities. But it does work with quadratic curvature - it weakens high-energy interactions and allows for convergence.
Cons: Creates "ghosts" (particles with negative energies or negative probabilities, which create their own problems). There's various proposed solutions, but none that's really a "eureka!" moment. Generally along the lines of "they exist but are purely virtual and don't interact", "they exist but they're so massive that they decay before they can interact with the universe", "they don't exist, we're just using the math out of bounds and need a different representation of the same", "If we don't stop at R^2 but also add in R^3, R^4, ... on to infinity, then they go away". Etc.
The theory isn't new, BTW. The idea is from 1918 (just a few years after Einstein's theory of General Relativity was published), and the work that led to the "Pros" above is from 1977.