En majo kaj junio ni pli ofte sidas sur la balkono, dum la infanoj ludas en la ĝardeno.

Because en is the normal preposition for in when talking about months, seasons, and many other time expressions.

• en majo = in May
• en junio = in June

So en majo kaj junio means in May and June.

Esperanto usually does not use la with month names in this kind of general time expression.

In Esperanto, names of months are normally written with lowercase letters, unlike in English.

So:

• majo
• junio

not Majo or Junio, unless the word begins the sentence.

ofte means often.
pli means more.

So pli ofte means more often.

You only need ol when you explicitly say what the comparison is:

• pli ofte ol vintre = more often than in winter

In your sentence, the comparison is understood from context, so ol is not necessary.

sidi means to sit or to be sitting.

So ni sidas can mean:

• we sit
• we are sitting

The Esperanto present tense often covers both the simple present and the English present progressive, depending on context.

Here sidas works well because it describes a repeated or usual action in that period.

These are three related but different verbs:

• sidi = to be sitting, to sit
• sidiĝi = to sit down, to become seated
• sidigi = to seat someone, to make someone sit

Examples:

• Mi sidas. = I am sitting.
• Mi sidiĝas. = I sit down.
• Ŝi sidigas la infanon. = She seats the child.

In your sentence, sidas is correct because it describes the state or activity of sitting.

Yes. Both are possible.

Esperanto word order is fairly flexible, and adverbs like pli ofte can move around a bit.

• Ni pli ofte sidas sur la balkono
• Ni sidas sur la balkono pli ofte
• Pli ofte ni sidas sur la balkono

All are understandable. The version in your sentence puts a little emphasis on pli ofte before the verb.

Because a balcony is normally thought of as a surface or platform that you are on, not something you are inside.

So:

• sur la balkono = on the balcony

Using en would suggest being inside something enclosed, which is not how balconies are usually viewed.

Here dum means while.

It connects two actions that happen at the same time:

• ni pli ofte sidas sur la balkono
• dum la infanoj ludas en la ĝardeno

So the idea is: one action is happening while the other is happening.

dum can also mean during when followed by a noun phrase:

• dum la somero = during the summer

Because these are understood as specific, identifiable things:

• la balkono = the balcony
• la infanoj = the children
• la ĝardeno = the garden

Esperanto uses la much like English the.

So this sentence sounds like it is talking about:

• a particular balcony
• particular children
• a particular garden

There is no indefinite article in Esperanto, so there is no word for a/an.

Because these words are part of location phrases, not direction phrases.

• sur la balkono = on the balcony
• en la ĝardeno = in the garden

There is no accusative -n here because nobody is moving onto or into those places; they are already there.

Compare:

• Ni sidas sur la balkono. = We are sitting on the balcony.
• Ni iras sur la balkonon. = We go onto the balcony.
• La infanoj ludas en la ĝardeno. = The children are playing in the garden.
• La infanoj kuras en la ĝardenon. = The children run into the garden.

Because en la ĝardeno tells you where the children are playing.

The verb ludi here simply means to play, and the place is given with a prepositional phrase:

• ludas en la ĝardeno = play in the garden

If you want to say what game they are playing, then ludi can take a direct object:

• ludi ŝakon = to play chess
• ludi futbalon = to play football/soccer

But in your sentence, the important information is the location, not the game.

• ĝ is pronounced like the j in job
• j is pronounced like the y in yes

So:

• ĝardeno sounds roughly like jar-DE-no
• junio sounds roughly like yu-NI-o

Also remember that Esperanto stress normally falls on the second-to-last syllable:

• ĝarDEno
• juNIo