Ŝi promesas, ke ŝi mem purigos la pladojn, se mi kuros al la butiko por ovoj.

Ke means that and introduces a clause that gives the content of the promise.

• Ŝi promesas = She promises
• ke ŝi mem purigos la pladojn = that she herself will clean the dishes

In English, that is often optional: She promises (that) she will clean...
In Esperanto, ke is normally stated.

Because the clause after ke needs its own subject.

• Ŝi promesas — subject: ŝi
• ke ŝi mem purigos la pladojn — subject again: ŝi

So Esperanto says explicitly who will do the cleaning. English does the same here: She promises that she will clean...

Mem adds emphasis. Here ŝi mem means she herself.

It does not mainly mean alone. It means that she, and not someone else, will do it.

So:

• ŝi purigos la pladojn = she will clean the dishes
• ŝi mem purigos la pladojn = she herself will clean the dishes

Purigos is the future-tense form of purigi.

You can break it down like this:

• pur- = clean / pure
• -ig- = make, cause to become
• -os = will

So purigos literally means will make clean. In normal English, that is just will clean.

The -n marks the direct object.

Here, la pladojn is the thing being cleaned, so it takes the accusative ending:

• ŝi purigos la pladojn = she will clean the dishes

This is one of the most important uses of -n in Esperanto.

Because por is a preposition, and nouns after ordinary prepositions do not usually take -n.

So:

• por ovoj = for eggs

The -n ending is not used here because ovoj is not the direct object of the verb. It belongs to the prepositional phrase por ovoj.

Esperanto normally uses the tense that matches the actual time meant.

Since the running would happen in the future, Esperanto uses the future tense:

• se mi kuros = literally if I will run

But in natural English, we usually say if I run rather than if I will run. So this is a place where Esperanto and English differ.

Yes. Kuros is the future of kuri, so it literally means will run.

• mi kuros al la butiko = I will run to the store

Depending on context, English might translate this more naturally as I’ll run to the store or I’ll dash to the store, meaning a quick trip.

Al means to, toward and is used for movement in the direction of something.

So:

• al la butiko = to the store

This shows motion toward the store. If there were no movement, Esperanto would use a different expression.

La is the definite article, meaning the.

• la pladojn = the dishes — a specific set of dishes
• ovoj = eggs / some eggs — not specified as a particular known set

Esperanto has no separate word for a/an, so leaving out la often gives an indefinite sense.

Because Esperanto normally separates subordinate clauses with commas.

So the sentence is divided like this:

• Ŝi promesas, ke ŝi mem purigos la pladojn, se mi kuros al la butiko por ovoj.

The comma before ke marks the that-clause, and the comma before se marks the if-clause. This is more regular in Esperanto than in English.

Ŝi is pronounced roughly like shee in English.

• ŝ sounds like English sh
• i sounds like ee

So ŝi = shee.