The ideas of dark matter and dark energy are mind blowing. Why is it said that dark matter overcomes dark energy in galaxies but it loses the battle in intergalactic space? In other words, why is dark energy dominant between galaxies but not inside galaxies?
That's simply because there the dark matter density in galaxies is high, but the density in intergalactic space is low. The reason for that, in turn, is because dark energy is a property of spacetime itself - any spacetime, wherever it is. In contrast, dark matter like normal matter follows gravitational forces and clumps into galaxies and other such structures.
These aspects of astronomy and cosmology are indeed very interesting and very significant, but don't allow the names to get in the way of your understanding. Dark matter is a form of matter made (most likely) of particles which don't interact very much with the matter we are more familiar with (i.e. protons, neutrons, electrons etc.). The evidence for it has several strands (rotation curves of galaxies, gravitational lensing, calculations of structure formation, calculations of matter content from nucleosynthesis in the early universe, etc.)
The evidence for dark energy is summarised here: What is the evidence that dark energy exists? (as of 2020))
"Dark energy" is a rather confusing name, in my opinion. It refers to the behaviour of the expansion of the universe at the largest scales. Ordinary matter tends to pull things together by gravitational attraction and therefore always slows the expansion. But the equations of general relativity allow that there might be effects which accelerate the expansion. Such effects get the name "dark energy". I wish the cosmologists had settled on a better name. But there it is. The name arises because this contribution to the overall dynamics of the universe enters the equations in two places, one of which behaves like energy and the other of which behaves like stress, in fact a form of tension (the opposite of pressure). But in physics if something behaves like X then we say it is X. So it is called energy. Dark because it does not emit electromagnetic radiation.
The most significant thing about this contribution called dark energy is that it enters the equations of general relativity as a term which just gets added on, irrespective of where the matter in the universe may be. It is added on in exactly the same way everywhere. And most of the universe is vast empty voids between filaments of dark matter. Therefore the dark energy contribution adds up to a large total effect on average, even though it is tiny compared to the ordinary matter and dark matter at any given place where matter is present. The reason why the gravitational attraction of ordinary matter and of dark matter easily wins against the repulsive effects of this other term, wherever the matter is actually present, is simply that the dark energy per unit volume is so small. But after averaging over the whole volume of the universe it nevertheless makes the biggest contribution to the dynamics of the whole universe on average, because it is present throughout the otherwise empty voids, and those voids make up most of the volume.