A quick reminder of what density estimates are and why they are useful. Note that with the usual definition of $\psi(x)=\sum_{p^n \le x}\log p$ where $p$ denotes a prime, we have that $$\psi(x)-x=\sum_{|\Im \rho| \le T}\frac{x^{\rho}}{\rho}+O(\frac{x\log^2 x}{T}), 2 \le T \le x$$ where the sum is on the nontrivial zeroes of $\zeta$ up to height $T$. In particular for $h<x$ we have:$$|\psi(x+h)-\psi(x)|=h-\sum_{|\Im \rho| \le T}\frac{(x+h)^{\rho}-x^{\rho}}{\rho}+O(\frac{x\log^2 x}{T}), 2 \le T \le x$$
So if we prove that for some particular $h$ the error terms above (and $\sqrt x$ which comes from the squares of primes in the definition of $\psi$) is smaller than $h$ then we get there is a prime in $(x, x+h]$ and of course, the error above is smallest, if the real parts of the zeroes are smallest possible and that is precisely RH with all $\Re \rho =1/2$ which gives that $h>c\sqrt x \log^2 x$ is allowable so we get primes in $(x, x+x^{1/2+\epsilon})$ say for all $\epsilon >0$ and large enough $x$
However we do not know RH, but it turns out that by clever integration, one can average the sums on zeroes so even if we have zeroes with larger real part (so RH is false and the real parts are greater than $1/2$ though of course still less than $1$), what counts in this particular problem of getting smaller and smaller $h$ for which we have primes in $(x, x+h]$ is the number of zeroes with larger real part and we denote that by $$N(\sigma, T)=\sum_{\Re \rho \ge \sigma, \Im \rho \ge 0}1$$
In particular, RH is clearly equivalent to $N(\sigma, T) =0$ for all $\sigma>1/2$ and all $T>0$.
It also turns out that we can get good results for the above problem, comparable to the ones obtained on RH, when we only have $$N(\sigma,T) << T^{2(1-\sigma)+\epsilon}, 1/2\le \sigma \le 1$$ for every $\epsilon>0$ and this is called the Density Hypothesis (DH).
It is known (and not hard to prove) that the Lindelof Hypothesis (LH) which gives bounds on the size of $\zeta$ on vertical lines (and is equivalent to $N(\sigma, T+1)-N(\sigma, T)=o(\log T), \sigma >1/2$ so is implied of course by RH which has all those terms zero) implies the density hypothesis, so we have the chain $RH \implies LH \implies DH$. Since we are very very far from useful attacks on RH or LH (notwithstanding the usual and the new crank claims), anything that improves bounds regarding DH is considered a major breakthrough.
Until now the best uniform bound $$N(\sigma, T)<< T^{12(1-\sigma)/5+\epsilon}$$ was the Huxley one (better bounds depending on $\sigma$ in a more complicated way are known is some ranges but not in the whole $(1/2,1)$) which allowed an $h=x^{7/12}+\epsilon$ in the above ($7/12=1-5/12$).
Maynard and Guth proved the improved bound $$N(\sigma, T)<< T^{30(1-\sigma)/13+\epsilon}$$ by improving the Huxley-like bound in some ranges and using the above better bounds in other ranges, so getting to an allowed $h=x^{17/30+\epsilon}$ above ($17/30=1-13/30$)
Note that as noted the optimal exponent in DH is $2$ (due to the fact that $N(1/2,T)>> T\log T$) which leads to the best $h$ being comparable to the one on RH so $x$ exponent $(1-1/2)+\epsilon=1/2+\epsilon$.
However, as the above mentions, DH is considerably weaker than LH (and hence RH) and the newer breakthrough only improves slightly towards DH (from $2.4$ to $2.307$ when we want $2$) hence while of course a major ANT breakthrough, it is unclear if it leads anywhere regarding RH per se
