There are three things with different meanings here

• Tomber de l’escalier → Falling from the stairs
• Tomber sur l’escalier → Falling on the stairs
• Tomber dans l’escalier

The first two are the same in English and in French. Now for the third one: when one walks up or down the stairs, one is in the stairs, as one would be in a field or in a house. So when one falls while being in the stairs, it makes sense to say that one fell in the stairs — tombe dans l’escalier.

In a way, tomber dans l’escalier is tomber de l’escalier dans l’escalier, falling in the stairs from the stairs.

Note that it is not exactly the same as falling down the stairs, since it doesn’t imply that the fall brought the faller all the way down the stairs, that would be tomber jusqu’en bas de l’escalier.