What's to explain? In root concept, at least, the idea has been around since the early 18th Century - some argument may be made its origins predate even that by some considerable margin, but, as technology was as yet unequal to the task of providing pertinent data, none of that is particularly important. Early in the 20th Century, however, it was determined that, from the perspective of an Earth-bound observer, then-as-yet not entirely understood cosmologic objects were receeding from Earth at enormous velocity. At around the same time, Einstein developed the Theory of General Relativity, which, despite his fervent wishes and ardent efforts to the contrary, does not allow for a "Steady State" or "Eternally Constant" universe. Much intellectual energy, involving the best minds of the times, went into attempts to salvage the "Steady State" universe, but to no avail.
In the late 1920s, the first postulate describing what eventually has come to be known as "The Big Bang" was proposed by LemaƮtre, a Belgian cleric and accomplished, respected astronomer/physicist/theoretician, who largely, though not solely, developed the idea from Einstein's Theory of General Relativity with strong influence coming also from the work of Russian astronomer/physicist Freidman. Within a relatively short period of time, what is known as the Friedmann-LemaƮtre-Robertson-Walker Metric provided an exact solution to Einstein's field equations of general relativity - though the solution allowed equally well for either a dynamic (expanding and/or contracting) universe or a static (Steady State) universe. Toward the end of the 1920s, the pace of discovery really began to accellerate, and Hubble's work all but put paid to "Steady State" theory - though at the time and for decades to follow, even Hubble himself sought desperately to prove otherwise.
In the late 1940's, Gamow (a driving force in the development of quantum theory, among others such as Fermi, Teller, Bethe, and Oppenheimer), Russian by birth but by then a naturalized US citizen, published an enormously influential paper decribing the observed levels and distribution of hydrogen and helium throughout the universe as being wholly consistent with circumstances which only could have been consequent to something very like the Big Bang. Gamow in that paper also predicted there should be a detectable, relatively uniform "afterglow" from the proposed Big Bang, in the form of infrared radiation a few degrees above Absolute Zero. Still, the "Big Bang"/"Steady State" debate went on; despite mounting evidence for, and the discovery of absolutely no evidence counter to The Big Bang Theory, Steady State had its staunch and well-credentialled champions.
In the mid 1960's, Penzias and Wilson's discovery of Cosmic Background Radiation confirmed Gamow's prediction; "Steady State" was dead, and Expansion was the deal. Over the past 4 decades or so, the case for the Big Bang/Expansion theory has grown exponentially, while, though open questions remain, no practical, feasible alternative has been presented. Discoveries and confirmations over just the few years yet accomplished in the 21st Century have served only to push remaining uncertainties into a steadily shrinking corner.
In the sense that a scientific theory is the functional equivalent of a repeatedly tested, constantly and ever-increasingly amplified and confirmed hypothesis consistent with all other relevant laws, principles, and theories, without substantive counter evidence, the Big Bang more than makes the cut. The math works, confirming the conclusions drawn from both physical and experimental observations, and the observations, physical and experimental, confirm the math.
In sum, and quite simply, there at present is no viable competition for The Big Bang Theory, and by all indication, it appears highly unlikely, to the point of near absolute statistical certainty, that ever will there be. The more it is studied and tested, the more we learn, the stronger The Big Bang Theory becomes.
Ellis, G and van Elst, H (1999):
Cosmologic Models (note: 90 page .pdf download) provides a thorough and elegant (though not-really-for-the-layman) explanation of why The Big Bang Theory is the best currently available theory, and in what particulars it is superior to any prior, alternate or competing theory - in short, why and how it is the only theory which works.
Less scientifically precise but far more accessible is this Wikipedia article:
History of the Big Bang