Kiam mia kolegino komencis ŝerci, mia humoro tuj fariĝis pli bona.

Kiam means when. Here it introduces a time clause:

• Kiam mia kolegino komencis ŝerci = When my colleague started joking

So it tells us at what time the main action happened.
In Esperanto, kiam can be used both:

• in questions: Kiam vi venos? = When will you come?
• in subordinate clauses: Kiam li alvenis, ni manĝis. = When he arrived, we ate.

In your sentence, it is the second use.

Because the part beginning with kiam is a subordinate clause, and it is separated from the main clause by a comma:

• Kiam mia kolegino komencis ŝerci, = subordinate clause
• mia humoro tuj fariĝis pli bona. = main clause

This is very normal in Esperanto.

The suffix -in- marks female sex in Esperanto.

So:

• kolego = colleague
• kolegino = female colleague

The word is built like this:

• koleg- = colleague root
• -in- = female
• -o = noun ending

So kolegino literally means female colleague.

In Esperanto, possessive words like mia, via, lia, and so on usually make the noun definite already.

So Esperanto normally says:

• mia kolegino
• mia humoro

not:

• la mia kolegino
• la mia humoro

Also, Esperanto has no indefinite article like a or an in English.

This is a very common Esperanto pattern:

• komenci
  • infinitive

So:

• komencis = began / started
• ŝerci = to joke

Together:

• komencis ŝerci = started joking or began to joke

The second verb stays in the infinitive form -i after komenci.

Because -i is the infinitive ending, the form meaning to do something.

After verbs like komenci, voli, povi, devi, Esperanto often uses an infinitive:

• Mi volas dormi. = I want to sleep.
• Ŝi povas veni. = She can come.
• Ili komencis ridi. = They started laughing.

So komencis ŝerci is perfectly regular Esperanto.

Ŝerci means to joke or to make jokes.

So mia kolegino komencis ŝerci means she began speaking or acting jokingly. It does not necessarily mean she told one specific joke; it can mean she started joking around in general.

Tuj means immediately, right away, or at once.

Here it modifies fariĝis:

• mia humoro tuj fariĝis pli bona = my mood immediately became better

Its position is fairly flexible, but this placement is natural and clear. It emphasizes that the mood changed quickly.

Fariĝis means became.

It comes from:

• fari = to do / to make
• fariĝi = to become

The ending -iĝ- often gives the idea become, get into a state, or undergo a change.

So:

• mia humoro fariĝis pli bona = my mood became better

This shows a change of state, not just a description.

Because fariĝis shows change, while estis just describes a state.

Compare:

• Mia humoro estis pli bona. = My mood was better.
• Mia humoro fariĝis pli bona. = My mood became better.

In your sentence, the idea is that the colleague’s joking caused or coincided with an improvement in mood, so fariĝis is the better choice.

Because bona describes the noun humoro.

• humoro = a noun
• words describing a noun must be adjectives
• adjective ending = -a

So:

• pli bona = better as an adjective

If you used bone, that would be an adverb, used to describe a verb or an entire action:

• Mi fartas pli bone. = I feel better.

But here we are describing the mood, so pli bona is correct.

Because there is no direct object here.

• mia kolegino is the subject of komencis
• mia humoro is the subject of fariĝis
• pli bona is a predicate adjective describing humoro

The accusative -n is usually used for a direct object or sometimes for motion toward something. Neither applies here.

Because it agrees with humoro, which is singular.

In Esperanto, adjectives agree with the nouns they describe in:

• number
• accusative marking, if any

So:

• humoro = singular
• bona = singular adjective

If the noun were plural, you would get:

• humoroj
• pli bonaj

Yes, it could, but the nuance is slightly different.

• komencis ŝerci = started joking
• ekŝercis = began joking / burst into joking

The prefix ek- often marks a sudden beginning of an action.

So ekŝercis would be possible, but komencis ŝerci is very straightforward and easy to understand.

A very literal breakdown is:

• Kiam = when
• mia kolegino = my female colleague
• komencis ŝerci = started to joke
• mia humoro = my mood
• tuj = immediately
• fariĝis = became
• pli bona = better

So a literal version is:

When my female colleague started to joke, my mood immediately became better.

A more natural English translation would often be:

When my colleague started joking, my mood immediately improved.