Бабушка села на табуретку у окна и начала чистить яблоки.

Because села means sat down — it focuses on the change into a seated position.

• сесть / села = to sit down
• сидеть / сидела = to be sitting

So:

• Бабушка села на табуретку = Grandma sat down on the stool
• Бабушка сидела на табуретке = Grandma was sitting on the stool

This is a very common Russian distinction: one verb for the movement/change, another for the state.

That ending shows past tense feminine singular.

The subject is бабушка, which is grammatically feminine, so past-tense verbs agree with it:

• сел = masculine
• села = feminine
• село = neuter
• сели = plural

So:

• дедушка сел
• бабушка села

The same applies to начала.

Because села involves movement onto something, so Russian uses на + accusative.

• на табуретку = onto the stool
• на табуретке = on the stool (location)

Compare:

• Она села на табуретку. = She sat down on the stool.
• Она сидела на табуретке. = She was sitting on the stool.

This is a very important pattern:

• куда? where to? → often accusative
• где? where? → often prepositional (or another location case, depending on the preposition)

Табуретку is accusative singular.

The dictionary form is табуретка. Since it is a feminine noun ending in -а, the accusative singular usually changes -а to -у:

• табуретка → nominative
• табуретку → accusative

That is why after села на... you get на табуретку.

У окна means by the window, near the window, or next to the window.

The preposition у requires the genitive case, so:

• окно = nominative
• окна = genitive singular

So:

• у окна = by the window
• literally something like at the window’s side / near the window

This is a very common expression in Russian.

Russian often uses начать + infinitive to mean to begin/start doing something.

So:

• начала = began
• чистить = to peel / clean

Together:

• начала чистить яблоки = began to peel the apples

This is a normal structure, just like English began to....

Because after начать, Russian usually uses an infinitive that presents the action as a process.

Чистить is imperfective and means the ongoing activity of peeling/cleaning.

That fits well with начала:

• начала чистить яблоки = she started the process of peeling apples

Using a perfective infinitive here would usually sound unnatural or would express a different nuance.

In this context, it most naturally means peel apples.

The verb чистить can mean clean, but with food it often means peel, skin, or prepare by removing the outer part:

• чистить картошку = peel potatoes
• чистить яблоки = peel apples

So even though the basic idea is clean, the natural English translation here is usually peel.

Here яблоки is accusative plural.

The dictionary form is яблоко. Its nominative plural is яблоки, and for inanimate nouns, the accusative plural is usually the same as the nominative plural.

So:

• nominative plural: яблоки
• accusative plural: яблоки

Since apples are the direct object of чистить, accusative is required.

Because in Russian, just like in English, if the subject stays the same, you normally do not repeat it.

So:

• Бабушка села на табуретку у окна и начала чистить яблоки.

means:

• Grandma sat down on the stool by the window and began peeling apples.

The subject of both verbs is clearly бабушка, so there is no need to repeat it.

This word order is quite natural and neutral for narration.

• Бабушка = the topic/subject
• села на табуретку у окна = first action
• и начала чистить яблоки = second action

Russian word order is more flexible than English, but changing it changes emphasis.

For example:

• У окна бабушка села на табуретку... puts more focus on by the window
• Яблоки бабушка начала чистить... puts more focus on the apples

The original sentence sounds like normal storytelling.

It usually means grandmother or grandma, but in some contexts it can also refer to an elderly woman.

In a sentence like this, without more context, most learners will understand it as Grandma.

So depending on context, бабушка can be:

• your grandmother
• someone else’s grandmother
• an old woman, in a colloquial or familiar way

But grandma is the most likely reading here.

Then Russian would probably use imperfective verbs describing an ongoing scene:

Бабушка сидела на табуретке у окна и чистила яблоки.

Notice the changes:

• села → сидела
  from sat down to was sitting
• на табуретку → на табуретке
  from motion onto to location on
• начала чистить → чистила
  from began peeling to simply was peeling / peeled

This is a great example of how Russian uses different verbs and cases to distinguish action beginning from ongoing state/action.