La tinkturo ŝanĝis la koloron de la punto.

The -n marks the direct object: the thing that was changed.

• la tinkturo = the subject, the thing doing the action
• ŝanĝis = changed
• la koloron = the direct object, the thing affected by the action

So koloro becomes koloron because the dye changed the color.

Also notice:

• la does not change
• punto does not get -n here, because it is inside the de phrase de la punto, not the direct object itself

De usually means of and shows a relationship between nouns.

Here, de la punto tells you whose or which color it is:

• la koloron de la punto = the color of the lace

English often uses of or an apostrophe, but Esperanto commonly uses de for this kind of relationship.

So this phrase is not a second object. It just modifies koloron.

Because -is is the Esperanto past-tense ending.

The verb pattern is very regular:

• ŝanĝas = changes / is changing
• ŝanĝis = changed
• ŝanĝos = will change
• ŝanĝi = to change

So ŝanĝis tells you the action happened in the past.

Esperanto uses la for the, and it can appear before each definite noun phrase.

Here there are two definite noun phrases:

• La tinkturo = the dye
• la koloron de la punto = the color of the lace

So both get la.

Also, Esperanto has no separate word for a/an. If you want to say a dye or a color, you often just use the noun without any article.

Yes, to some extent. Esperanto word order is fairly flexible because endings show grammatical roles.

The most neutral order here is:

• La tinkturo ŝanĝis la koloron de la punto.

Because koloron has -n, you could rearrange the sentence and still keep the meaning clear, for example:

• La koloron de la punto ŝanĝis la tinkturo.

That is grammatical, but less neutral and more marked in style. For learners, the standard subject-verb-object order is usually best.

In this sentence, ŝanĝi is being used transitively: to change something.

So:

• La tinkturo ŝanĝis la koloron = The dye changed the color.

If you want to say that the color changed by itself, Esperanto often uses ŝanĝiĝi:

• La koloro de la punto ŝanĝiĝis. = The color of the lace changed.

That -iĝ- idea often means to become or to undergo a change.

Yes, punto means lace.

This is a useful vocabulary point because it can surprise English speakers. If you want point, Esperanto usually uses punkto.

So:

• punto = lace
• punkto = point, dot, period/full stop

They are different words, and mixing them up is easy at first.

The two special letters here are:

• ŝ = like sh in ship
• ĝ = like j in judge

So ŝanĝis is pronounced roughly like SHAHN-jis.

A few more pronunciation notes:

• Esperanto stress is always on the second-to-last syllable.
• ŝanĝis has two syllables: ŝAN-ĝis
• tinkturo is tink-TU-ro
• koloron is ko-LO-ron
• punto is PUN-to

That regular stress system makes pronunciation much easier once you get used to it.