Cuando llego a casa, me pongo la bata y me siento en el sillón un rato.

Because this sentence describes a habit or routine: Whenever I get home, I put on my robe and sit in the armchair for a while.

In Spanish, for repeated actions like this, cuando + present tense is normal:

• Cuando llego a casa, ... = When/Whenever I get home, ...

If you were talking about one future occasion, Spanish often uses cuando + present subjunctive:

• Cuando llegue a casa, me pondré la bata.
  = When I get home, I’ll put on my robe.

So:

• cuando llego = habitual / repeated
• cuando llegue = future event

A casa is a very common expression meaning home.

• llegar a casa = to get home
• estar en casa = to be at home
• irse a casa = to go home

When Spanish means home in this general, personal sense, it usually does not use an article.

A la casa usually means to the house as a physical building, not necessarily home.

Compare:

• Llego a casa. = I get home.
• Llego a la casa de Ana. = I arrive at Ana’s house.

Me pongo comes from ponerse, which often means to put on clothing.

So:

• me pongo la bata = I put on the robe

The me is there because the action is done to yourself. Spanish often uses reflexive verbs for things like getting dressed:

• me pongo la chaqueta = I put on my jacket
• me quito los zapatos = I take off my shoes

This is very natural Spanish.

In Spanish, with clothing and body parts, the definite article is often used instead of a possessive like my, especially when it is already obvious who it belongs to.

So Spanish prefers:

• me pongo la bata
• literally: I put on the robe
• natural meaning: I put on my robe

Because of me, it is already clear that it’s your own robe.

This also happens a lot with body parts:

• me lavo las manos = I wash my hands
• not usually me lavo mis manos

Here, bata most likely means a robe, dressing gown, or housecoat.

In Spain, bata can mean different things depending on context, for example:

• a robe/dressing gown worn at home
• a lab coat
• sometimes a kind of smock or apron

Because the sentence says when I get home and then sit in the armchair, the home meaning is the most likely one: robe/dressing gown.

Good question: me siento can come from two different verbs:

1. sentarse = to sit down
2. sentirse = to feel

So me siento could mean either:

• I sit down
• I feel

Here, the context makes it clear that it means I sit down, because of en el sillón:

• me siento en el sillón = I sit down in the armchair

If it meant I feel, you would usually expect something after it like:

• me siento cansado = I feel tired
• me siento bien = I feel well

Because sentarse normally goes with en when talking about the place where you sit:

• sentarse en una silla = to sit in/on a chair
• sentarse en el sofá = to sit on the sofa
• sentarse en el sillón = to sit in the armchair

So:

• me siento en el sillón = I sit down in the armchair

Also, sillón is not just any chair. It usually means an armchair or easy chair, something bigger and more comfortable than a simple silla.

• silla = chair
• sillón = armchair / upholstered chair / easy chair

A sillón is usually larger and more comfortable, often used in a living room.

So me siento en el sillón un rato gives a cozy image: sitting in an armchair for a while.

Un rato means for a while or for a little while.

It refers to an unspecified short period of time.

Examples:

• Descanso un rato. = I rest for a while.
• Voy a leer un rato. = I’m going to read for a while.

In your sentence:

• me siento en el sillón un rato = I sit in the armchair for a while

Because Spanish often omits subject pronouns when they are not needed.

The verb ending already shows the subject:

• llego = I arrive / I get
• pongo = I put
• siento = I sit

So yo is unnecessary unless you want emphasis or contrast.

Compare:

• Cuando llego a casa... = neutral, normal
• Cuando yo llego a casa... = more emphasis, maybe contrast with someone else

Because when a verb is conjugated, reflexive pronouns usually go before it.

So:

• me pongo
• me siento
• me lavo

If the verb were in the infinitive or a command, the pronoun could attach to the end:

• voy a ponerme la bata
• ponte la bata
• sentarse en el sillón

But in your sentence the verbs are conjugated in the present tense, so the normal position is:

• me pongo
• me siento

Because Spanish uses the present tense very often for habitual actions, just like English does.

This sentence is basically saying:

• Whenever I get home, I put on my robe and sit in the armchair for a while.

So the present tense is not about what is happening right now only. It can also describe:

• habits
• routines
• general truths

That is exactly what is happening here.

Because Cuando llego a casa is an introductory time clause, and Spanish normally separates that kind of clause from the main clause with a comma when it comes first.

So this is standard:

• Cuando llego a casa, me pongo la bata...

If the order is reversed, the comma is often not used:

• Me pongo la bata y me siento en el sillón un rato cuando llego a casa.

So the comma here is mainly about sentence structure and readability.