Agárrate al pasamanos cuando bajes las escaleras, que el suelo sigue mojado.

Because the verb here is agarrarse a algo, which means to hold on to something. In this sentence, te is the reflexive pronoun for tú, and in an affirmative command it gets attached to the end:

• agarra = grab / take hold
• agárrate = hold on

The written accent is needed because adding -te changes the natural stress of the word, so Spanish adds the accent to keep the original pronunciation.

Because al is the contraction of a + el.

Here, agarrarse normally uses a:

• agarrarse a algo = to hold on to something

So:

• a el pasamanos becomes al pasamanos

This contraction is mandatory in standard Spanish.

Yes, here pasamanos is singular: el pasamanos = the handrail.

Some Spanish nouns end in -s even in the singular. The article tells you the number:

• el pasamanos = the handrail
• los pasamanos = the handrails

So even though it ends in -s, it is singular in this sentence.

Because after cuando, Spanish often uses the present subjunctive when referring to a future action.

Here the meaning is basically when you go down the stairs / when you next go down the stairs, so the action has not happened yet from the speaker’s point of view. That is why Spanish uses:

• cuando bajes

If you said cuando bajas, it would usually sound more like a habitual action:

• cuando bajas las escaleras = when/whenever you go down the stairs

So bajes is the normal choice for a warning or instruction about a future moment.

Because in Spanish, verbs like subir and bajar can directly take something like las escaleras as their object:

• subir las escaleras = to go up the stairs
• bajar las escaleras = to go down the stairs

This is very natural Spanish. English often uses a different structure, but in Spanish no extra preposition is necessary here.

You can also hear bajar por las escaleras, but that slightly emphasizes the route or means of going down. In this sentence, bajar las escaleras is perfectly normal.

Here que means something like because or since.

So:

• ..., que el suelo sigue mojado
  = ..., because the floor is still wet

This is a very common structure in spoken Spanish and informal written Spanish. It gives the reason for what was just said.

A more explicit alternative would be:

• Agárrate al pasamanos cuando bajes las escaleras, porque el suelo sigue mojado.

Both are correct, but que sounds very natural and conversational.

Because seguir + adjective means to remain or to still be.

So:

• está mojado = it is wet
• sigue mojado = it is still wet / it remains wet

The sentence is emphasizing that the floor was wet before and it has not dried yet. That is why sigue mojado is a better fit here.

Because mojado agrees with el suelo, which is masculine singular.

In Spanish, adjectives agree in gender and number with the noun they describe:

• el suelo mojado
• la escalera mojada
• las escaleras mojadas

So here mojado matches suelo.

Yes. You could say:

• Agárrate al pasamanos cuando bajes las escaleras, porque el suelo sigue mojado.

That would also be correct.

The difference is mostly one of style and tone:

• que = very common, conversational, natural in speech
• porque = a bit more explicit and neutral

In everyday Spanish, especially in speech, que is extremely common in this kind of sentence.

In Spain, if you are speaking to vosotros, you would say:

• Agarraos al pasamanos cuando bajéis las escaleras, que el suelo sigue mojado.

The main changes are:

• agárrate → agarraos
• bajes → bajéis

A useful detail: with affirmative vosotros commands of reflexive verbs, the final -d of the imperative drops before -os:

• agarrad + os → agarraos

That is the standard Spain form.