Li pli bone iras, ekde kiam ŝi komencis masaĝi lian kokson.

Both are possible. Esperanto word order is fairly flexible, and pli bone iras puts a little more focus on better.

• Li iras pli bone = He walks/goes better.
• Li pli bone iras = He walks/goes better, with better slightly emphasized.

A learner will probably hear Li iras pli bone more often because it feels more neutral.

Literally, iras means goes. But in context, it can naturally mean walks or gets around.

Because the sentence mentions massaging his hip, Li pli bone iras most likely means:

• he walks better
• he is moving around better

So this is a good example of Esperanto using a general verb where English might prefer a more specific one.

Because bone is an adverb, and it modifies the verb iras.

• bona = good, as an adjective
• bone = well, as an adverb

Here we are describing how he goes/walks, so we need the adverb:

• Li iras bone = He walks/goes well.
• Li iras pli bone = He walks/goes better.

If you used bona, it would need to describe a noun, not the verb.

Ekde kiam means since or more literally starting from the time when.

Breakdown:

• ekde = since, from
• kiam = when

So:

• ekde kiam ŝi komencis masaĝi lian kokson = since she began massaging his hip

It marks the starting point of a situation that continues up to now.

Yes, many speakers would also say de kiam.

• de kiam = since
• ekde kiam = since, emphasizing the starting point a bit more

So both are understandable:

• Li pli bone iras, de kiam ŝi komencis masaĝi lian kokson.
• Li pli bone iras, ekde kiam ŝi komencis masaĝi lian kokson.

Ekde can feel slightly more explicit about from that point onward.

Komencis masaĝi means began to massage.

This is a very common Esperanto structure:

• komenci + infinitive
• ŝi komencis masaĝi = she began to massage

You could also form a verb with ek-:

• ŝi ekmasaĝis lian kokson = she started massaging his hip

Both are possible, but komencis masaĝi is very straightforward and often easier for learners to understand.

Because lian kokson is the direct object of masaĝi.

• li = he
• lia = his
• lian = his, accusative form because it belongs to a direct object
• kokso = hip
• kokson = hip, direct object

So:

• masaĝi lian kokson = to massage his hip

In Esperanto, when a noun takes -n as a direct object, any adjective or correlating word with it also agrees. Here lia becomes lian.

Kokso usually means hip.

Depending on context, English might translate it as:

• hip
• hip joint
• side of the pelvis

In this sentence, his hip is the most natural translation.

Because the sentence describes a present result:

• Li pli bone iras = He walks better now / He is walking better now

Then the second part gives the starting point:

• ekde kiam ŝi komencis... = since she began...

So the pattern is:

• present situation now
• past event that started the change

This is very similar to English:

• He walks better since she began massaging his hip is awkward in English
• more natural English would be He has been walking better since she began massaging his hip

Esperanto often uses the simple present in places where English may prefer a present perfect construction.

Because the beginning happened in the past.

• ŝi komencis masaĝi = she began to massage
• ŝi komencas masaĝi = she is beginning to massage / begins to massage

The sentence says his improved walking started from that earlier point, so komencis is the correct tense.

No. It means she began massaging his hip, but it does not by itself say whether she did it once or repeatedly.

It simply marks the start of that action or practice. The context may suggest repeated treatment, but the grammar alone does not force that meaning.

Yes, depending on context.

• masaĝi lian kokson literally = to massage his hip

In natural English, you might say:

• massage his hip
• work on his hip
• give his hip a massage

But the Esperanto verb itself is just the straightforward to massage.

Because ŝi is the subject of komencis, while lian kokson is the object of masaĝi.

• ŝi komencis = she began
  
  • ŝi is the subject, so no -n
• masaĝi lian kokson = to massage his hip
  
  • kokson is the direct object, so it gets -n
  • lian agrees with kokson, so it also gets -n

This is a very common feature of Esperanto agreement.

Not usually from this sentence alone. Iras strongly suggests movement, going, or walking.

If you wanted He is doing better in a more general sense, Esperanto would probably use a different expression, such as something built around:

• farti = to fare, to be doing
• senti sin = to feel

So here the most natural understanding is specifically about his walking or mobility.

It is normal and helpful. The comma separates the main clause from the time clause:

• Li pli bone iras, ekde kiam ŝi komencis masaĝi lian kokson.

Esperanto punctuation is not always identical to English punctuation, but this comma is perfectly natural because it marks a clear break between the statement and the explanation of when it became true.