Nach dem Regen könnte man auf den nassen Stufen leicht ausrutschen, deshalb halte ich mich am Geländer fest.

Because nach takes the dative case in this kind of time expression.

• nach dem Regen = after the rain
• dem is the dative form of der Regen

So this is not about movement; it is simply the fixed preposition nach followed by the dative.

No. German man is an impersonal pronoun, not the noun man meaning a male person.

Here, man means something like:

• one
• you
• people
• someone in general

So könnte man ... ausrutschen means one could slip or more naturally in English, you could easily slip.

Könnte is the Konjunktiv II form of können. In a sentence like this, it often makes the statement sound a little more cautious, hypothetical, or general.

Compare:

• kann man ausrutschen = you can slip
• könnte man ausrutschen = you could slip

In this sentence, könnte fits the idea of a possible risk after the rain, rather than a flat statement of fact.

It is dative plural.

The preposition auf is a two-way preposition, so it can take:

• accusative for direction/movement toward something
• dative for location

Here the meaning is on the wet steps, meaning location, so German uses the dative.

That gives:

• den = dative plural article
• nassen = adjective ending used after a definite article in dative plural
• Stufen = plural noun; it already ends in -n, so there is no extra visible dative ending to add

So:

• auf den nassen Stufen = on the wet steps

Because könnte is a modal verb.

In a main clause with a modal verb:

• the conjugated modal verb goes in the usual second position
• the main verb in the infinitive goes to the end

So:

• könnte = second position
• ausrutschen = clause-final infinitive

This is a very common German pattern:

• Ich kann kommen.
• Man könnte ausrutschen.

Here leicht means easily.

So leicht ausrutschen means to slip easily or to be likely to slip.

It does not mean lightly in this sentence. It describes how easy it would be for someone to slip on the wet stairs.

Because the sentence contains two main clauses:

1. Nach dem Regen könnte man auf den nassen Stufen leicht ausrutschen
2. deshalb halte ich mich am Geländer fest

Deshalb is not a conjunction like weil or und. It is an adverb meaning therefore / that’s why / for that reason.

So the comma separates the two main clauses. A semicolon or full stop would also be possible.

Because German main clauses follow the verb-second rule.

That means the finite verb must come in the second position. If deshalb takes the first position, the verb must come next:

• Deshalb halte ich mich am Geländer fest.

So the structure is:

1. deshalb
2. halte
3. ich
4. rest of the clause

This is called inversion from an English learner’s point of view, but in German it is just the normal verb-second pattern.

Because the expression is sich an etwas festhalten or sich festhalten.

So in this meaning, festhalten is used reflexively:

• ich halte mich fest
• literally: I hold myself firmly
• natural meaning: I hold on

That is why the reflexive pronoun mich is needed.

Because festhalten is a separable verb in this use.

In a normal main clause, the conjugated part stays near the front, and the separable prefix goes to the end:

• Ich halte mich fest.
• Ich halte mich am Geländer fest.

But when the verb is in the infinitive, it stays together:

• Ich will mich am Geländer festhalten.

So in your sentence:

• halte = conjugated verb
• fest = separated prefix at the end

Am is the contraction of an dem.

• an dem Geländer → am Geländer

This is very common in German.

Here Geländer is in the dative because the phrase is an + dative in this context:

• sich am Geländer festhalten = to hold on to the railing

So am Geländer is the normal compact form of an dem Geländer.

Yes. German is flexible about what comes first in a main clause. You could say:

• Ich halte mich deshalb am Geländer fest.

That is also correct.

The version you have puts the reason first in the first clause, then uses deshalb to introduce the result. It sounds natural and clear. German often moves time expressions like Nach dem Regen to the front for emphasis or flow.