As far as I know is the discovery that galaxies that are farther away are moving faster from us than galaxies the are closer by. This led to the theory of Dark Energy. The acceleration was speeding up. However, is the explanation not much simpler? The farther we look back the more juvenile the galaxies are. The Big Bang happened for those galaxies not as long ago compared to galaxies closer by. If the Big Bang happened 13.8 billion years ago and if we are looking at a galaxy which is 10 billion light years away than for that particular Galaxy the Big Bang happened only 3.8 billion years ago (13.8 - 10). During the Big Bang the inflation happened faster at the beginning than later, hence a ordinary explosion. Right after the explosion started, the material - in our case the galaxies - are moving faster. Gravity will eventually slow it down. In more nearby galaxies, gravity had much more time to slow the acceleration down because the Big Bang happened longer ago than for more distant galaxies as we see them today. What am I missing in this?
You are not missing anything, apart from a slight misinterpretation of the type Ia supernova cosmology results.
With no dark energy, the rate of expansion of the Universe would slow down. The gravitational "pull" of everything would be responsible for this de-celeration.
When we look at a distant galaxy, we see it as it was when the light was emitted. Indeed, more distant galaxies have two components to their redshift: if the expansion of the universe was neither accelerating or decelerating, Hubble's law, that the recession velocity was proportional to distance, would apply at all distances.
But in addition, for a decelerating universe, the recession velocity would be higher than predicted by Hubble's law - or to put it another way, the value of the Hubble "constant" was bigger in the past.
Now the type Ia supernova cosmology experiment set out to test this by measuring the velocities and effectively measuring the distance to very distant galaxies. You can try to match these velocity, distance pairs to an expanding universe model that has a time-varying Hubble constant. What they found was that at some point in the past, the deceleration of the universe stopped and it started to accelerate. The velocities of the distant galaxies was not high enough to match a purely decelerating universe.
The conclusion reached, now supported by other evidence - notably the cosmic microwave background fluctuations - is that there is a form of energy density in the universe (a pressure if you like), that causes space to expand, and in the latter billions of years of our Universe's development, this "dark energy" has come to dominate its dynamics, causing the expansion to accelerate.
I like the following image (from Perlmutter et al. 2003) - it has been populated with more data now - but it shows the point. The curves show the scale factor of the Universe as a function of time. A uniform expansion (a universe with no dark energy or gravitating matter would just be a straight line). The gradient of this graph at any point in time tells you the value of the Hubble "constant" at that time. The Universes with dark energy have a point of inflection where the gradient stops decreasing and starts increasing - i.e. the universe decelerates for a bit and then starts to accelerate. The observations of type Ia supernovae looked far enough back that they went (just) beyond this point of inflection and were able to see the difference between the measured velocities and the velocities predicted by a purely decelerating universe. There are quite a lot more measured supernovae now populating the region around 10 billion years ago, but I can't immediately lay my hands on an updated version.