What is the scientific explanation that suggests the universe is expanding? What two main concepts does it rely on?

The theory that the universe is expanding began with one important observation made by Edwin Hubble: red shift. In 1929, Hubble stated that all galaxies were moving away from us and from each other. He based this on the observation that the light being emitted from the galaxies was "shifted" toward the red end of our visible light spectrum.  You experience a somewhat similar effect when the frequency of sound appears higher as an object approaches you and lower as it recedes away (commonly known as the Doppler Shift).  Light being emitted from the galaxies is being "stretched" as the galaxies move away, increasing their wavelength and decreasing their frequency.  We perceive this as a shift towards red (as it has a longer wavelength than the other colors).

What becomes more interesting is that if one is to assume that the galaxies are all moving apart from each other, then they must have been closer together in the past.  This is an essential concept for the Big Bang Theory--all matter and energy in the universe was compressed into a tiny point approximately 13.8 billion years ago.  It then "exploded" outward--though thinking of it as an explosion as we understand it doesn't actually illustrate what happened.  Space is literally flowing into existence between matter pockets.

What's even more interesting is that the relative level of red shift increases the further away you look.  One would assume that since gravity would be the only force acting on distant galaxies, they should be slowing down, ultimately coming to a stop, and hurling back together (proposed as a Big Crunch).  The exact opposite has been observed--galaxies are accelerating away from each other (and the universe is increasing in size at an increasing rate).  Since we have no force in our four fundamental force model (gravity, EM, strong, and weak nuclear), scientists have proposed something unseen is acting in the space between galaxies--dark energy. 

So one would have to argue that dark energy is the most compelling explanation of why the universe is expanding.