More ELI12 but here we go...

We have a theory called 'General Relativity' (GR), first proposed by Einstein in the early 20th century. It describes how matter and energy distort space-time. In this theory, energy can lead to weird features in this space-time depending on your position in space ( you can travel through ripples, bumps, sinks, wells...) or on your 'position' in time (space-time can oscillate or expand or shrink even if you remain at a given place). Our everyday experience of this behavior of space-time is called 'gravity'.

Now using telescopes, we are able to see effects of gravity in the Universe. For instance we see that the distances in the Universe are not static : the Universe is said to be currently in 'expansion'. GR has no problem to explain that : in fact it predicts that if you have a certain amount of energy in a given volume, space-time cannot remain static and usually it expands, driven by this density of energy.

Moreover, not only do we see that it is expanding, but we have a very good idea of the history of expansion of the Universe, of the way the expansion evolved with time. For instance we observe that the expansion was slower and slower during the first half of the Universe history and became faster and faster during the second half.

Interestingly, General Relativity has no problem to explain this two eras but it requires two different kinds of energy.

The first epoch (when expansion is slower and slower) is driven by the energy stored in matter : matter is conserved but becomes more and more diluted as the Universe expands. The energy density of matter drives the expansion during this first era but is less and less efficient in doing so, because matter is less and less dense due to this same expansion.

The second epoch is also easily explained by GR as soon as you have a type of energy which is not diluted as the Universe expands. If it's not diluted by growing distances, this kind of energy is always efficient at driving the expansion and it can accelerate as time goes by.

If we try to reproduce accurately the observed history of the Universe expansion, we end up with an energy budget of ~30% for matter and ~70% for the second kind of energy not diluted by expansion. Note that if you have a type of energy that is diluted as time goes by (the matter in our case), it is expected that at some point it will be dominated by another kind of non-diluted energy, if the latter exists : you just have to wait long enough for that, roughly 8 billions years in our case. This is the reason why matter is driving the expansion at early times and not at latter time.

But what is this second type of non-diluted energy ? Well, first it's not that complicated to imagine a form of energy that is not diluted as distance grows in the Universe : we just have to create more energy as distances, areas or volumes increases. For instance let's imagine that any given volume must come with a certain amount of energy, for example a constant quantity energy for every 'gallon/liter' of space : if distances grow, your 'gallon/liter' will become more 'gallons/liters' and the amount of energy will increase as well. This kind of energy is therefore non-diluted by an expanding Universe. We call it 'Dark Energy' because we don't know what it is, but it is also named 'cosmological constant' (because this irreducible amount of energy that comes with every volume does not depend on your position or instant) or 'vacuum energy' (because even if you consider a volume devoid of matter, there comes this energy).

Admittedly, it's weird to have this kind of Dark Energy BUT if we believe that General Relativity is the ultimate description of gravity and space-time, Dark Energy must exists to explain an history of expansion that cannot be driven by matter only and it must constitute 70% of the energy budget of the Universe. If General Relativity is not the ultimate description of gravity, it's another story...

PS : obviously, I have taken shortcuts and made some approximations. Also I haven't explained why a large fraction of matter is said to be dark and made of non-ordinary matter. But that's enough for the moment, maybe later...