Hace frío fuera, así que prefiero quedarme dentro.

Because hace frío is the normal Spanish weather expression for it’s cold.

Spanish often uses hacer with weather and temperature:

• hace frío = it’s cold
• hace calor = it’s hot
• hace viento = it’s windy
• hace sol = it’s sunny

By contrast:

• es frío usually describes something or someone as cold by nature
• está frío usually means something is cold at the moment

So for the weather outside, hace frío is the natural choice.

Spanish does not use a dummy subject like English it in weather expressions.

English says:

• It is cold
• It is raining

Spanish just says:

• Hace frío
• Llueve

So hace frío is an impersonal expression: there is no real subject written or implied.

Because Spanish often leaves out subject pronouns when the verb ending already shows who is doing the action.

Here, prefiero clearly means I prefer because the -o ending marks yo.

So:

• prefiero = I prefer
• yo prefiero = I prefer

Both are possible, but yo is usually only added for emphasis, contrast, or clarity.

For example:

• Prefiero quedarme dentro. = neutral
• Yo prefiero quedarme dentro. = emphasizes I

Because prefiero is the main verb of the sentence, and the sentence needs a conjugated verb.

After verbs like preferir, Spanish often uses an infinitive for the second action when the subject is the same:

• Prefiero quedarme dentro. = I prefer to stay inside.
• Quiero salir. = I want to go out.
• Necesito descansar. = I need to rest.

So the pattern is:

preferir + infinitive

Here, the same person does both actions:

• I prefer
• I stay

That is why Spanish uses quedarme, not another conjugated form.

Because quedarse means to stay or to remain, and that is the meaning needed here.

The verb quedar by itself has several other meanings, such as:

• to remain / be left
• to fit
• to arrange to meet

But quedarse is the common verb for staying somewhere.

Examples:

• Me quedo en casa. = I’m staying at home.
• Prefiero quedarme dentro. = I prefer to stay inside.

So the -se version is important here.

Because me is the reflexive pronoun that goes with quedarse, and when a reflexive verb is in the infinitive, the pronoun is attached to the end.

So:

• quedarse = to stay
• quedarme = to stay myself / for me to stay

That is the normal pattern:

• lavarme = to wash myself
• sentarme = to sit down
• quedarme = to stay

After preferir, you say:

• prefiero quedarme

Not:

• prefiero me quedar

Así que means so, therefore, or so that means. It introduces a result or consequence.

In this sentence:

• Hace frío fuera = reason / situation
• así que prefiero quedarme dentro = result

So the logic is: It’s cold outside, so I prefer to stay inside.

They are related, but they do different jobs.

• así que = so, introduces the result
• porque = because, introduces the reason

Compare:

• Hace frío fuera, así que prefiero quedarme dentro.
  It’s cold outside, so I prefer to stay inside.

• Prefiero quedarme dentro porque hace frío fuera.
  I prefer to stay inside because it’s cold outside.

Same basic idea, but the direction of the explanation changes.

Yes. Here they are adverbs, so they can stand alone.

• fuera = outside
• dentro = inside

You only need de when they are followed by a noun:

• dentro de la casa = inside the house
• fuera del edificio = outside the building

But when no noun follows, dentro and fuera are enough:

• Quédate dentro. = Stay inside.
• Está fuera. = He/She is outside.

They often mean the same thing:

• fuera / afuera = outside
• dentro / adentro = inside

In Spain, fuera and dentro are especially common.
Afuera and adentro also exist, but they may sound more common in many Latin American varieties or slightly more colloquial, depending on the region.

So in Spanish from Spain, fuera and dentro are a very natural choice.

Yes, but the nuance changes a little.

• prefiero quedarme dentro = I prefer to stay inside
  Emphasizes not going out

• prefiero estar dentro = I prefer to be inside
  Focuses more on the state or location

• prefiero quedarme en casa = I prefer to stay at home
  More specific: inside where? At home

So the original sentence is natural if you want the general idea of staying indoors rather than going outside.