Morcovii fierb repede, iar după aceea îi amestec cu cartofii.

Because they are definite plural forms.

For these masculine nouns:

• morcov = carrot
• morcovi = carrots
• morcovii = the carrots

And:

• cartof = potato
• cartofi = potatoes
• cartofii = the potatoes

So the two i letters are not accidental:

• one i belongs to the plural form
• the other i is the definite article

Because the subject is plural: morcovii = the carrots.

The verb a fierbe in the present tense has:

• el/ea fierbe = he/she/it boils
• ei/ele fierb = they boil

So:

• Morcovul fierbe = The carrot boils
• Morcovii fierb = The carrots boil

Here a fierbe means to boil / to cook by boiling.

This verb can be used in two ways:

• intransitive: something boils / cooks
  
  • Morcovii fierb = The carrots are boiling / cook
• transitive: someone boils something
  
  • Fierb morcovii = I am boiling the carrots

In your sentence, it is the first type: the carrots are the thing being cooked.

Iar is a conjunction that often means and, while, whereas, or sometimes gives a slight sense of moving on to the next part.

In this sentence, it is very natural as:

• and
• and then

So:

• Morcovii fierb repede, iar după aceea...

means something like:

• The carrots cook quickly, and after that...

Compared with și, iar often sounds a little more like a transition from one clause to the next.

După aceea is a common Romanian expression meaning after that or afterwards.

It is written as two words because it is a phrase, not a single fused word.

You could also hear:

• apoi = then
• după asta = after that
• pe urmă = afterwards / later

All of these can work, but după aceea is a very normal standard choice.

Îi refers back to morcovii.

It is a direct object pronoun meaning them for a masculine plural noun.

So:

• îi amestec cu cartofii = I mix them with the potatoes

Here, them = the carrots.

Because Romanian unstressed object pronouns normally come before the verb.

So Romanian says:

• îi amestec = I mix them
• îl văd = I see him/it
• o cunosc = I know her

This is different from English, where object pronouns usually come after the verb.

So îi amestec is normal Romanian word order, while amestec îi would not be standard here.

Because Romanian often leaves subject pronouns out when the verb ending already makes the subject clear.

Amestec already means:

• I mix

So eu is not necessary.

You would add eu only for emphasis or contrast:

• Eu amestec, nu tu = I mix, not you

Cu cartofii means with the potatoes.

Romanian often uses the definite form when the things are already known from the situation, especially in practical contexts like cooking. The potatoes are understood as the potatoes involved in this dish.

So:

• cu cartofii = with the potatoes
• cu cartofi = with potatoes / with some potatoes

Both can exist in Romanian, but the definite form sounds very natural when the ingredients are already specific in context.

They are similar-looking spellings, but they are not exactly the same.

• -ii in morcovii, cartofii is the ending of the word and is usually pronounced as a clear final ee sound, often slightly lengthened or with two vocalic elements depending on speech.
• îi begins with Romanian î, a central vowel that English does not really have, followed by a short i glide.

For learners, the main thing to notice is:

• morcovii and cartofii end with -ii
• îi is a separate pronoun and starts with î

Even if fast speech makes them subtle, Romanian speakers hear them as different forms.