Tintagel, I've kind of already mentioned the following but here is how I would answer the question if it were asked to me about Dark Energy:
1. Newton notices that things fall and as such we call the force that causes this gravity.
2. He makes gravitational laws which seem infallible.
3. Astronomy improves and we notice anomalies in space which seem to contradict laws of inertia concerning the acceleration of galaxies.
4. Knowing gravitational laws still hold true locally, DE is theorized to account for the strange behavior of accelerating galaxies.
5. Evidence pops up that supports DE:
a. Supernovae are useful for cosmology because they are excellent standard candles across cosmological distances. They allow the expansion history of the Universe to be measured by looking at the relationship between the distance to an object and its redshift, which gives how fast it is receding from us. The relationship is roughly linear, according to Hubble's law.
Recent observations of supernovae are consistent with a universe made up 71.3% of dark energy and 27.4% of a combination of dark matter and baryonic matter.
b. Measurements of cosmic microwave background (CMB) anisotropies indicate that the universe is close to flat. For the shape of the universe to be flat, the mass/energy density of the universe must be equal to the critical density. The total amount of matter in the universe (including baryons and dark matter), as measured from the CMB spectrum, accounts for only about 30% of the critical density. This implies the existence of an additional form of energy to account for the remaining 70%.
c. The theory of large-scale structure, which governs the formation of structures in the universe (stars, quasars, galaxies and galaxy groups and clusters), also suggests that the density of matter in the universe is only 30% of the critical density.
d. Accelerated cosmic expansion causes gravitational potential wells and hills to flatten as photons pass through them, producing cold spots and hot spots on the CMB aligned with vast supervoids and superclusters. This so-called late-time Integrated Sachs-Wolfe effect (ISW) is a direct signal of dark energy in a flat universe.