Mi abuela se agarra al pasamanos cuando baja las escaleras.

Because in this sentence the natural verb is agarrarse a + thing, which means to hold on to something.

• agarrar algo = to grab something
• agarrarse a algo = to grab onto / hold on to something

So:

• Mi abuela agarra el pasamanos = my grandmother grabs the handrail
• Mi abuela se agarra al pasamanos = my grandmother holds on to the handrail

The se here is part of the verb pattern, not a separate word you can just remove without changing the meaning.

Al is the contraction of a + el.

• a el pasamanos → al pasamanos

Here it appears because agarrarse is used with a:

• agarrarse a algo = to hold on to something

So se agarra al pasamanos literally means she holds herself to the handrail, but in normal English that is just she holds on to the handrail.

Because the verb pattern requires the preposition a:

• agarrarse a
  • noun

So you cannot normally say:

• se agarra el pasamanos

if you mean holds on to the handrail.

That would sound more like she grabs the handrail as a direct object, rather than the more idiomatic holds on to it.

Pasamanos means handrail or banister rail.

It is usually masculine:

• el pasamanos
• un pasamanos

A useful thing to notice is that although it ends in -s, it can still be singular:

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

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

Here:

• baja = she goes down / comes down

Since the sentence already starts with Mi abuela, there is no need to add ella.

Spanish does this all the time:

• Mi abuela se agarra al pasamanos cuando baja las escaleras.
• not usually Mi abuela se agarra al pasamanos cuando ella baja las escaleras.

Adding ella is possible, but it would usually be for emphasis or contrast.

Both can be possible, but bajar las escaleras is a very common and natural way to say to go down the stairs.

So:

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

You may also hear:

• bajar por las escaleras

which emphasizes the route a bit more, something like go down via the stairs. But in everyday Spanish, bajar las escaleras is extremely common.

Because Spanish very often uses escaleras in the plural to mean stairs or a staircase.

Compare:

• la escalera can mean the ladder or the staircase as a structure
• las escaleras often means the stairs

So bajar las escaleras is the usual everyday expression for going down the stairs.

The present tense is being used for a habitual action or something that generally happens.

• Mi abuela se agarra al pasamanos cuando baja las escaleras.

This means something like:

• My grandmother holds on to the handrail when she goes down the stairs
• My grandmother holds on to the handrail whenever she goes down the stairs

So it is not necessarily happening right now. It describes a usual behavior.

Because here cuando refers to something habitual or repeated, so Spanish uses the indicative:

• cuando baja las escaleras = when she goes down the stairs / whenever she goes down the stairs

If you were talking about a future event that has not happened yet, Spanish often uses the subjunctive:

• Cuando baje las escaleras, la ayudaré.
• When she goes down the stairs, I’ll help her.

So:

• cuando baja = habitual / usual / factual
• cuando baje = future or not-yet-realized event

On its own, agarrar often does mean to grab or to take hold of. But in agarrarse al pasamanos, the meaning is softer and more natural in English as hold on to.

So although the basic idea is still taking hold, the best translation here is usually:

• holds on to the handrail

rather than:

• grabs the handrail

because English grab can sound more sudden or forceful.

Yes, agarrarse al pasamanos is understandable and natural in Spain.

Other common options are:

• sujetarse al pasamanos = to hold on to the handrail
• cogerse al pasamanos = to hold on to the handrail
• apoyarse en el pasamanos = to lean on the handrail

The nuance is slightly different:

• agarrarse = physically take hold of
• sujetarse = hold oneself steady by holding
• apoyarse = lean against / use for support

So the original sentence is perfectly good, especially if the idea is that she grips the rail for safety.

Because mi already functions as a determiner.

In Spanish, you normally do not use an article with a possessive like mi, tu, su, etc.

So:

• mi abuela = my grandmother

not:

• la mi abuela in standard modern Spanish

Yes. Bajar can work in more than one way.

It can be intransitive:

• Mi abuela baja lentamente. = My grandmother goes down slowly.

And it can also be used with things like las escaleras:

• bajar las escaleras = to go down the stairs

This is very normal Spanish. So in your sentence, las escaleras is the thing being gone down, even though in English we usually need down + the stairs.

Yes, that is also correct.

• cuando baja las escaleras = when she goes down the stairs
• al bajar las escaleras = when going down the stairs / on going down the stairs

The version with cuando is often a little more straightforward for learners. The version with al + infinitive is very common too and can sound slightly more compact.