Actually, the first thing gleaned from the
COBE microwave background radiation map was that
Earth is actually moving through the universe.
The map showed a slightly higher temperature on one half-sphere of the sky than on the other, due to the doppler effect. From this, scientists could conclude not only that Earth was moving through the universe, but also that it was moving in that direction and at a certain speed relative to the background radiation.
After filtering out this strong temperature difference, another, smaller, movement was found. This one, iirc, was due to Earth's movement through the solar system, while the former was a result of sol's movement around the milky way.
Only now could the irregularities described in the above node be seen.
Another strong piece of evidence supporting the big bang is using the second law of thermodynamics in reverse. It says that the entropy in the universe never decreases with time, which means that, going back through time, it must decrease, due to e.g. friction. Therefore, at some point in the past, the universe must have been in a state of total order (no entropy) and extreme heat. This could supposedly only be achieved if the universe's volume was very very small. Sounds like the start of a big bang, don't you think?