Ŝi diras, ke varma teo helpas ŝin trankviliĝi post longa tago kun multe da streso.

Ke means that and introduces a subordinate clause:

Ŝi diras, ke... = She says that...

In Esperanto, ke is usually kept. English often drops that, but Esperanto much more often says ke explicitly. So for a learner, it is safest and most natural to include it.

Because they are adjectives. In Esperanto, adjectives normally end in -a:

• varma = warm
• longa = long

They also agree with the noun in number and case. Here both nouns are singular and have no accusative -n, so the adjectives stay varma and longa.

Because varma teo is the subject of helpas.

In this clause:

varma teo helpas ŝin trankviliĝi

the tea is doing the helping, so it is the subject. Subjects do not take -n.

The direct object is ŝin, so that is the word with -n.

Because helpi takes a direct object: you help someone.

So:

• ŝi = she, subject form
• ŝin = her, object form

In varma teo helpas ŝin trankviliĝi, the tea helps her, so ŝin is correct.

Trankviliĝi means to become calm, to calm down, or to grow calm.

It comes from:

• trankvila = calm
• -iĝ- = become, get into a state

So trankviliĝi is not exactly to calm someone else, but rather to become calm oneself.

Because -ig- and -iĝ- mean different things:

• trankviligi = to make someone calm, to calm someone
• trankviliĝi = to become calm, to calm down

This sentence says that warm tea helps her calm down, so trankviliĝi fits well.

If you said varma teo trankviligas ŝin, that would mean warm tea calms her more directly. The original sentence is slightly different: the tea helps her get calm.

Because Esperanto tense usually follows the actual meaning, not automatic English-style tense shifting.

So if the idea is still true now, you can say:

• Ŝi diras, ke varma teo helpas ŝin... = She says that warm tea helps her...

Even after a past reporting verb, Esperanto often keeps the tense that matches the real situation:

• Ŝi diris, ke varma teo helpas ŝin...
  She said that warm tea helps her.

That can sound unusual to English speakers, because English often shifts to helped more easily.

Because post is a preposition, and prepositions normally take a noun without accusative -n.

So:

• post longa tago = after a long day

The adjective matches the noun, so it is also longa, not longan.

English speakers sometimes expect -n for time expressions, but here the preposition post already shows the relationship, so no accusative is needed.

Multe da means a lot of.

This is a very common quantity pattern in Esperanto:

• multe da akvo = a lot of water
• multe da laboro = a lot of work
• multe da streso = a lot of stress

After da, the following noun names what there is a quantity of. Here streso is treated like a mass noun, so singular is natural.

This is why you do not get forms like multaj stresoj here unless you really mean separate individual stresses, which is not the usual idea.

Kun means with.

Here it describes tago:

post longa tago kun multe da streso
= after a long day with a lot of stress

So the idea is not that the tea helps her calm down with stress, but that she is calming down after a stressful day.

Yes, Esperanto word order is fairly flexible because endings show the grammar.

Still, the original order is the most neutral and natural:

Ŝi diras, ke varma teo helpas ŝin trankviliĝi post longa tago kun multe da streso.

You can move things for emphasis, but some versions sound less natural to learners' ears. For example, putting ŝin or post longa tago earlier can add emphasis, but the standard order is usually best unless you have a reason to highlight something.