Mi abuelo se agarra a la barandilla cuando baja las escaleras.

Because the natural pattern here is agarrarse a algo = to hold on to something.

• agarrar algo means to grab something
• agarrarse a algo means to grab onto / hold onto something

So se is part of the verb pattern, not a literal himself in English. In this sentence, mi abuelo se agarra a la barandilla means he holds onto the railing for support.

That a belongs to the verb pattern agarrarse a.

So:

• agarrarse a la barandilla
• agarrarse a una cuerda
• agarrarse a mi brazo

It is not the personal a. It is simply the preposition required after agarrarse when you say what someone is holding onto.

Spanish usually leaves out subject pronouns when they are clear from context.

Here, baja could mean he goes down, she goes down, or you go down formally, but the subject is already clear because of Mi abuelo. So Spanish does not need él.

This is very normal:

• Mi abuelo baja las escaleras
• Mi hermana trabaja mucho
• Juan vive en Madrid

Adding él is possible, but it would usually sound emphatic or contrastive.

Both can be correct, but they are slightly different in feel.

• baja las escaleras is a very common way to say he goes down the stairs
• baja por las escaleras puts more emphasis on the route: he goes down via the stairs

In everyday Spanish, bajar las escaleras is completely natural and very common. Spanish often uses verbs of movement this way.

Spanish often uses the definite article when the thing is already clear from the situation.

So las escaleras does not necessarily mean some special emphasis on the stairs. It usually just means the relevant stairs in that context, such as the stairs in his home or the stairs he is using.

This is very common in Spanish. English often uses possessives or leaves things more implicit, while Spanish often uses el/la/los/las.

Because the Spanish present tense is often used for habits and regular actions.

So Mi abuelo se agarra a la barandilla cuando baja las escaleras means this is what he usually does whenever that situation happens.

This is the same kind of present tense as in:

• Siempre desayuna temprano
• Trabajo en casa
• Cuando llueve, no salimos

It does not mean the action is happening right now.

In this kind of sentence, it can be understood as when in a general sense, which in English is often close to whenever.

Because the whole sentence describes a habit, cuando baja las escaleras means when / whenever he goes down the stairs.

So the idea is not one single occasion, but a repeated situation.

They are related, but not exactly identical.

• barandilla usually means the railing structure as a whole
• pasamanos is specifically the part you place your hand on, the handrail

In many everyday translations, both may end up as railing, banister, or handrail depending on context. In this sentence, barandilla is a very natural word for the railing by the stairs.

Yes. Sujetarse a la barandilla is also natural and means something like to hold onto the railing.

There are slight nuances:

• agarrarse suggests grabbing/holding on
• sujetarse suggests holding oneself steady or supporting oneself

Both work well here. Agarrarse sounds very natural in everyday speech.

Because bajarse is usually used for meanings like:

• to get off a bus, train, bike, etc.
• sometimes to come down from something
• in some contexts, to pull down / remove something from oneself

But for going down the stairs, the normal options are:

• baja las escaleras
• baja por las escaleras

So se baja las escaleras would not be the standard way to express this idea.

In some Spanish-speaking regions, yes, but for Spain Spanish, agarrarse a la barandilla is the safer and more standard choice.

So if you are learning Spanish from Spain, it is best to use:

• agarrarse a algo

rather than:

• agarrarse de algo

You may hear de in parts of Latin America, but a is what you should aim for here for Spain.