1 Lambert: Right then. We don't need horses and we don't need supplies. I guess we're on our way already.
2 Mordekai: Wait, I don't understand.
3 Lambert: Alvissa says there's plenty to eat in the Dentorn Forest.
4 Mordekai: No, I don't understand how we can be on our way without spending weeks getting into trouble in a village first.

Travelling long distances in a pre-industrial age was a thing that very few people did. It was just too difficult. To go somewhere more than a few miles away, you needed to make preparations and plan how you were going to cover the distance, how you were going to shelter and rest overnight, and how you were going to get food and water along the way.

Some typical movement rates (taken from a reliable reference source*) are about 24 miles per day on foot, or 32 miles per day on horse, assuming good terrain. These rates can vary a lot, depending on terrain, and how hard you push yourself or your horses. But assuming you're not riding your horses into the ground, their main advantage is to make the travel easier, not enormously faster.

So setting out without horses is not ridiculous. In case you thought there was anything silly about this comic.

* i.e. Dungeons & Dragons rules.