There are several independent strands of evidence that lead to that conclusion.
The first is the discovery, made by the astronomer Edwin Hubble by a series of measurements starting in 1924, is that distant galaxies recede away from us at a velocity that is linearly proportional to their distance; this is known as Hubble's law. The Belgian cosmologist Georges Lemaitre realized, in 1927 that the expanding universe can be tracked back in time to an originating single point.
Interestingly, Einstein's general theory of relativity, which was introduced already in 1915, 12 years earlier than Lemaitre's discovery, in fact allows (predicts) an expanding universe. However, since at 1915 the expansion was not yet discovered, and since Einstein himself believed that the universe is static, he added another parameter into his equations to allow for static universe. He threw away this parameter after Hubble's discovery that the universe in fact expands, but it made a magnificent re-appearance as "dark energy".
The second proof came about 40 years later, with the discovery of the cosmic microwave background radiation made in 1964 by Arno Penzias and Robert Wilson. The idea goes as follows. At very early times, the universe was so dense and hot that atoms could not have existed. Instead, the universe was composed by the basic ingredients of matter- electrons, protons and photons. In such a universe, photons were constantly scattered by the electrons, and could not travel far; very similar to what happens when you are inside a cloud, or driving in fog: you can't see very far.
As the universe expanded it cooled, and electrons and protons recombined to form atoms. At this stage, the photons could escape, and travel without being scattered. They travel since then. As the universe continues to expand and cools these photons are now very cold - having today a temperature of only 2.7 degrees above the absolute zero. Since they originate from the ancient universe, these photons are evenly distributed in all directions. This is the cosmic microwave background photons, first detected by Penzias and Wilson.
The third proof comes from measurements of the concentration of light elements in the universe (the so called "abundance of primordial elements"). These are light elements, such as Hydrogen, Helium and Lithium, which were formed shortly after the big bang, as the electrons and protons combined into atoms. It is possible to calculate the ratio of these elements, during the recombination process, and the calculation is in excellent agreements with observations.
The fourth proof is the evolution of galaxies over cosmic times. More distant galaxies are younger than close ones (since it takes light more time to travel to us from these galaxies), and so we can compare the morphology and distribution of galaxies over long times, to find that they indeed evolve – distant (young) galaxies look different than close (old) ones.
These are four independent measurements, that all lead to the same conclusion: the big bang certainly happened.