The evidence for Big Bang cosmology is an expanding universe. The expansion of the universe is accelerating. Gravity causes acceleration. What evidence is there that proves everything is moving away from us and not us falling away from everything towards a spacetime singularity?
If everything were accelerating towards a point, then this would impose an anisotropy in the redshift-distance relation of galaxies that depended on which direction they were in.
It is also not the case that the universal expansion has always accelerated. There is good evidence from observing high redshift $(z>1)$ type Ia supernovae that the universal expansion was decelerating up until a few billion years ago and then started to accelerate. This is what you expect if you add a dark energy component to the big bang model.
Finally, the expansion of the universe is only one element that is explained by the big bang model. Any alternative has to explain (at least): an almost isotropic cosmic microwave background; why the oldest objects appear to have particular abundances of helium and deuterium with respect to hydrogen; and how structure in the universe developed in a very particular way.
Bradyglennyeah asked: "What evidence is there that proves everything is moving away from us and not us falling away from everything towards a spacetime singularity?"
The observed homogenity and isotropy of the universe, in other words the fact that we see the same recession in every direction, while if everything was falling towards a singularity there would be a preferred direction towards everything was moving.
"Black hole cosmology", motivated by the similarity between the cosmological horizon and the event horizons of black holes, has been developed since Lee Smolin speculated (in a 1997 Oxford University Press book titled "The Life of the Cosmos") that the cosmos might've developed in a manner similar to biological evolution, with its changes likeliest to occur in regions where black holes were likeliest to form. Evidence for that possibility remains inconclusive, although much astrophysical evidence has substantiated the real existence of black holes: Since most stars are in binary pairs, it includes the nearly-circular orbits of stars isolated by the gravitational collapse of their former partner, and also includes the detection of gravitational waves consistent with collisions of black holes.
Smolin didn't elaborate much on whatever mechanisms might be involved, but Nikodem J. Poplawski has, since 2010, formulated a mechanism (described in many papers of his that can be seen for free on Cornell University's Arxiv website) which relies on the Einstein-Cartan Theory of gravity, itself developed in conversations between Einstein and the mathematician Elie Cartan in the late 1920's. Mathematically, ECT is more complicated than General Relativity, and requires that fermions (matter particles) should have some tiny spatial extent, rather than the "point-like" nature characterizing them in the conventional GR of 1915.
Here's how it works: When an extremely large rotating star (most stars rotate) has run out of its nuclear fuel, the lack of radiation pressure starts its gravitational collapse, and an event horizon begins propagating outward from the center of its volume. The horizon eventually separates many virtual particles (including fermions) from the antiparticles that had comprised pairs with them, with the fermions falling inward and the anti-fermions escaping. All fermions spin, and those in-falling fermions newly-materialized by the gravitational field, 32 orders of magnitude smaller than those which had comprised the star itself, eventually come into contact with the vastly larger stellar fermions, with that contact (the locally big "bang" within the hole) reversing their trajectories and greatly accelerating the velociy of their passage over them: They are spun outward into a region initially barely inboard of the former location of the star's surface, where they form a new "local universe" whose space will appear almost perfectly flat to its eventual inhabitants. Its subsequent expansion is more-or-less inertial.
Just as a cosmology based on the Oppenheimer-Snyder black holes would account for the radius/mass relationship mentioned by Safesphere, Poplawski's "torsion-based" inflationary cosmology may have one advantage over the slightly older theory of inflation based on a hypothetical scalar field: Because the formation of a black hole requires a star of larger mass than the variety of them that, upon exhaustion of their nuclear fuel, would collapse to form a neutron star, the addition of mass to a star nearing the time of gravitational collapse might result, instead, in the formation of a black hole containing a new local universe, which might (given the limited number of types of subatomic particles) itself rather closely resemble a downscaled version of the local universe inhabited by whatever beings would've decided to add that mass to it. (Poplawski's model might, in other words, allow a more extensive creative power to whatever biological or cybernetic beings might consequently decide to acquire additional mass for their star, as well as reducing the incoherent possibility that any such beings might lack functionality.)