How did ancient and medieval armies get their horses off their ships?

Great question!

One of the problems that had to be dealt with in medieval warfare was that you had all of these armies where the mounted knight dominated, but if you have to attack somewhere from the sea, you need to take your horses with you. 

And so a variety of strategies were used. For example, the Norman Invasion of 1066 involved a flotilla of longships which would land, and then horses would be led off the side of the ship and into the shallows/beach, as seen in the Bayeux tapestry:

And that works well as long as you control the beach-head and have time to get your horses off the ship, get them saddled and so forth, and then distribute them to their waiting riders. It’s not so good if your enemy are right there on the beach and able to disrupt your preparations. 

So by the time of the Fourth Crusade, we see specialized horse transports where knights could mount their horses on deck and charge straight off the side of the ship, presumably either via a ramp or gunwales that could be lowered: 

And they were quite successful, allowing the Crusaders to surprise the Byzantines with their speed and mobility and seize the vital fortress of Galata, which guarded the northern end of the boom chain that protected Constantinople’s Golden Horn. With the boom chain down, Venetian galleys could sail into the Golden Horn and launch amphibious attacks against the city’s less formidable northern walls.