Learn Hardcore Russian: Когда я спускаюсь по лестнице, я держусь за перила. - When I go down the stairs, I hold on to the handrail.

Because спускаться / спуститься means to go down / descend, and in Russian this verb is normally used with -ся when a person is descending.

я спускаюсь = I am going down / I go down
я спускаю usually means I lower something or I let something down

So:

Я спускаюсь по лестнице = I go down the stairs.
Я спускаю чемодан по лестнице = I lower/carry a suitcase down the stairs.

The -ся form shows that the subject is the one moving downward.

Why is it по лестнице and not на лестнице or в лестнице?

Russian uses по + dative to mean movement along, on, or down/up a surface or route.

So:

идти по улице = walk along the street
подниматься по лестнице = go up the stairs
спускаться по лестнице = go down the stairs

Here, по лестнице means along/down the staircase.

By contrast:

на лестнице = on the staircase, located there
в лестнице would usually not make sense here

So по is the normal preposition for this kind of movement.