There's really no way to answer this comprehensively, since the source base for the run from Marathon to Athens is extremely thin and doesn't offer any detailed context or justification. Instead, I'll just give a couple of points:
The story of the victory message does not occur in Herodotos' Histories, the closest thing we have to a contemporary source. Indeed, it takes more than 600 years to appear in the historic record, and most of the commonly known details of this story derive from the Roman author Lucian. According to Herodotos, the victorious Athenians marched back to Athens together as quick as they could, fearing that the Persian fleet would beat them to the city by sea. There's no good reason why they should have been preceded by a messenger except perhaps to warn the people in the city that the Persians were on their way. The story may just be a tall tale of endurance and self-sacrifice that never actually took place. This is not to say that we should therefore dismiss the episode as meaningless nonsense, but that we should bear in mind it is a story, and we can't claim to know better than what it tells us.
If we assume that there was a genuine run from Marathon to Athens, the route would have led through some very rugged ground. Herodotos tells us that the Persians chose to land at Marathon specifically because it was decent cavalry country; the rest of Attika was famously less hospitable to mounted troops. During the Peloponnesian War, constant sorties wore out the mounts of the Athenian cavalry very rapidly. It would have been far easier for a runner to leave the plain at Marathon and navigate the interior of Athenian territory than a horseman, even if he knew the way.
The story of the run from Marathon is often conflated with another story, which is recorded in Herodotos. This is the story of Pheidippides' run from Athens to Sparta (some 250km) to ask for reinforcements for the coming battle. This multi-day run is recorded because Pheidippides supposedly encountered the god Pan in the mountains of the Peloponnese, and Pan asked him why the Athenians did not offer worship to him. In response, the Athenians promised to build a shrine to Pan on the Akropolis, which Herodotos regards as an important reason for their victory.
Pheidippides is specifically identified by Herodotos as a "day-runner", which suggests that it was either his job or his personal specialism to run long distances. This has been taken by some to suggest an entire professional dynasty or clan at Athens who took care of sending messages to allies at speed. We don't have many other examples of the work of these long-distance runners, but they may have been Athens' poor counterpart to the well-developed Persian system of Royal Roads with horse relay stations, allowing messages to be delivered at speeds that baffled contemporary Greek observers like Herodotos.
There are many reasons why the Athenians would not have considered using horses for the job. The main one is that there were no public horses. All horses were privately owned, and they were enormously prestigious and prized possessions. While the state could compel rich citizens to serve as cavalry (and Athens had a cavalry of perhaps 100 men at the time of Marathon), they could not expect these wealthy horse lords to simply give up their horses or commit them to risky ventures like long-distance messaging. The likelihood of horses being ridden to exhaustion or lamed along the way would have made this an impossible proposition, not to mention the idea of simply leaving a horse that was worn out, or giving it away to be used by some messenger. Meanwhile, the absence of any Greek federal government structure or organisation prevented the establishment of a network of relay stations across the boundaries of city-states, even if states would have had the means to acquire and maintain horses for the purpose (which they did not, at least at the time of the Persian Wars). If we combine this with the natural aptitude of humans for long-distance running, it becomes clearer why the use of a mounted messenger may not have been obvious to the Athenians.
This is not to say that the Greeks were not aware of the possibility of using horsemen as messengers. In fact, they used horsemen to communicate between parts of the vast allied army gathered to fight the Persians at the decisive battle of Plataia in 479 BC. It's most likely that the horsemen used for this purpose by the Athenians were part of the 100 cavalry which already existed at the time of Marathon but are not recorded as a separate force at this earlier battle. Some scholars have suggested that all 100 horsemen were simply battlefield messengers and not "true" cavalry, but this seems excessive, and there was rarely any need for such messengers in Greek battles. Horsemen were probably used at Plataia because the distances were much shorter and the terrain (mostly) accomodated horses, meaning that the need for speed made them the better choice.