Он только вздохнул и сказал, что лучше бы нам встретиться завтра утром.

Он means he. In this sentence it is the subject of both verbs: вздохнул and сказал.

Russian often omits subject pronouns when the subject is already clear from context, but including он is perfectly normal, especially when introducing or reintroducing the person in the story.

Here только means something like just or only in the sense of all he did was...

So Он только вздохнул suggests:

• He just sighed
• He only sighed

The nuance is that he did not do anything more dramatic or elaborate first; he simply sighed and then spoke.

Вздохнул is:

• past tense
• masculine singular
• perfective aspect

It agrees with он, so the masculine ending makes sense.

The verb is вздохнуть — to give a sigh / to sigh once.
Because it is perfective, it presents the sigh as a single completed action: he sighed.

If it were feminine, it would be вздохнула.
If plural, вздохнули.

For the same reason as вздохнул: it agrees with он.

Russian past tense verbs change for gender and number:

• сказал — he said
• сказала — she said
• сказало — it said
• сказали — they said

So Он ... сказал is the normal masculine singular form.

There is no comma before и here because it connects two verbs with the same subject:

• Он только вздохнул и сказал...

Both actions belong to он, so this is a simple coordination inside one main clause.

There is a comma before что because что introduces a subordinate clause:

• сказал, что...
• said that...

That comma is required in standard Russian punctuation.

Что means that and introduces reported speech or a content clause after verbs like:

• сказать — to say
• думать — to think
• знать — to know
• понимать — to understand

So:

• сказал, что... = said that...

In English, that is often omitted, but in Russian что is normally kept.

Лучше бы is a very common pattern meaning something like:

• it would be better if...
• better to...
• it’d be best to...

In this sentence, лучше бы нам встретиться завтра утром means the speaker is proposing or recommending that meeting tomorrow morning would be preferable.

The particle бы adds a conditional or soft hypothetical nuance. Without бы, the sentence would usually sound more direct or practical:

• Нам лучше встретиться завтра утром = We’d better meet tomorrow morning
• Лучше бы нам встретиться завтра утром = It would be better for us to meet tomorrow morning

The version with бы can sound a little softer, more tentative, or more reflective.

Нам is the dative case of мы (we).

The construction here is impersonal:

• лучше бы нам встретиться

Literally, this is something like:

• it would be better for us to meet

Russian often uses the dative in this kind of structure to show the person for whom something is advisable, possible, necessary, etc.

Compare:

• Нам нужно идти — We need to go / literally It is necessary for us to go
• Мне пора — I have to go / It’s time for me
• Ему лучше остаться — He’d better stay

So нам is not the subject in the nominative; it is the person affected by the recommendation.

It is an infinitive because it depends on the лучше бы construction:

• лучше бы нам встретиться = it would be better for us to meet

Russian often uses an infinitive after words like лучше, надо, можно, нельзя, пора, etc.

It is perfective because the speaker means one completed meeting at a specific time.
Встретиться means to meet up / to have a meeting encounter once.

If you used встречаться, that would suggest repeated or ongoing meetings, which does not fit as well here.

So:

• встретиться = meet once, at a definite occasion
• встречаться = meet repeatedly / be meeting / date each other, depending on context

These are an aspect pair:

• встречаться — imperfective
• встретиться — perfective

In this sentence, the idea is a single future arrangement, so встретиться is natural.

Examples:

• Нам нужно встретиться завтра. — We need to meet tomorrow.
• Мы часто встречаемся по выходным. — We often meet on weekends.

Also, встречаться can mean to date:

• Они встречаются. — They are dating.

That meaning is not intended here.

Завтра утром means tomorrow morning.

Here:

• завтра = tomorrow
• утром = in the morning

Утром is the instrumental case of утро. Russian often uses the instrumental for times of day in adverbial expressions:

• утром — in the morning
• днём — in the daytime
• вечером — in the evening
• ночью — at night

So завтра утром is a standard time expression meaning tomorrow morning.

Yes. Russian word order is flexible, although each order gives a slightly different emphasis.

The sentence as written is neutral and natural:

• Он только вздохнул и сказал, что лучше бы нам встретиться завтра утром.

Possible alternatives:

• Он только вздохнул и сказал, что нам лучше бы встретиться завтра утром.
• Он только вздохнул и сказал, что завтра утром нам лучше бы встретиться.

These are also possible, but the emphasis shifts a little.
Russian often places the most informationally important part later, so завтра утром at the end sounds very natural.

Inside the subordinate clause, the main action is expressed by the infinitive встретиться, not by a finite verb.

Russian does not need a nominative subject there because нам already shows who is supposed to do the meeting:

• лучше бы нам встретиться
• literally: it would be better for us to meet

So the doer is understood from the dative pronoun нам.

Both can mean something like we’d better meet, but the nuance is a bit different.

• Нам лучше встретиться завтра утром
  More direct, practical recommendation: We’d better meet tomorrow morning.

• Лучше бы нам встретиться завтра утром
  Slightly softer, more hypothetical, sometimes more reflective: It would be better for us to meet tomorrow morning.

In real speech, both are common. The version with бы often sounds a little less blunt.

It is neutral and very natural. It would fit well in:

• narration
• conversation
• fiction
• everyday speech

Nothing in it is especially formal or slangy.
The phrase лучше бы нам встретиться завтра утром sounds like normal spoken Russian and also works well in written narrative.