Learn Hardcore Spanish (Latin America): Ella explica que una hipoteca grande sin buen salario puede ser un problema serio. - She explains that a big mortgage without a good salary can be a serious problem.

Explicar is the infinitive, meaning to explain.
In the sentence, we need she explains, which is the present tense, third person singular of explicar:

Yo explico – I explain
Tú explicas – You explain
Él / Ella / Usted explica – He / She / You (formal) explain

So Ella explica = She explains (right now / in general).

Do we really need to say Ella, or could it just be Explica que…?

In Spanish, the subject pronoun is often optional because the verb ending already shows who the subject is.

Ella explica que… – She explains that…
Explica que… – She explains that… (context must already make it clear who she is)

Both are grammatically correct.
In writing or in a longer conversation, speakers often drop ella if the subject is already obvious. Including Ella can be used to:

Introduce a new subject
Emphasize she (as opposed to someone else)

What does the que do in ella explica que…? Is it necessary?

Here que is a conjunction meaning that in English. It introduces a subordinate clause (what she explains):

Ella explica que… = She explains that…

In Spanish, this que is not optional.
You cannot say Ella explica una hipoteca grande… in this meaning; that would start to sound like She explains a big mortgage… (direct object) instead of She explains that a big mortgage… (reported speech / statement).

So in this structure:

explicar + que + clause = to explain that + clause

Why is it una hipoteca and not un hipoteca?

Because hipoteca is a feminine noun in Spanish:

la hipoteca – the mortgage
una hipoteca – a mortgage

Most nouns ending in -a are feminine, and hipoteca follows this rule.
The article (la / una) must agree in gender and number with the noun:

una hipoteca grande – a big mortgage (feminine singular)
unas hipotecas grandes – big mortgages (feminine plural)

Why is the adjective after the noun: una hipoteca grande, and not una grande hipoteca?

In Spanish, the default position of most adjectives is after the noun:

una hipoteca grande – a big/large mortgage

You can say una gran hipoteca, but:

gran (short form of grande) is used before the noun and often adds a sense of importance, greatness, or intensity, not just physical size.

So:

una hipoteca grande – a big mortgage (size/amount)
una gran hipoteca – a great / very important / huge (in impact) mortgage (more subjective or emphatic)

For a neutral, literal description, post-noun grande is the usual choice.

Why is it buen salario and not bueno salario?

Bueno has a shortened form (apócope) before a masculine singular noun:

bueno → buen before a masculine singular noun

So:

un salario bueno – a good salary (less common word order here)
un buen salario – a good salary (most natural)

Grammatically:

salario is masculine: el salario
Therefore: un buen salario, not un bueno salario

Why is there no un before buen salario? In English we say a good salary.

In Spanish, after sin (without), it is very common to omit the indefinite article when we talk about something in a general or non-specific way:

sin buen salario – without (a) good salary
sin dinero – without money
sin trabajo – without (a) job

You could say sin un buen salario, and it is also correct. The difference:

sin buen salario – more general, without good salary as a condition in life
sin un buen salario – sounds a bit more concrete, like without a good (enough) salary in a particular case

Both are acceptable; the version without un is very natural in this kind of general statement.

Why is it puede ser and not just es?

es un problema serio = it is a serious problem (stating it as a fact)
puede ser un problema serio = it can be / may be a serious problem (expressing possibility)

In the sentence, the idea is not that every large mortgage without a good salary is always a serious problem, but that it can become one or has the potential to be.

So puede ser (from poder, can / may) softens the statement and matches the English idea can be a serious problem.

Why not puede estar un problema serio?

Ser and estar both mean to be, but they’re used differently:

ser: essential characteristics, identity, classification
estar: temporary states, location, conditions

Problema serio = a type of problem, a characterization, so we use ser:

es / puede ser un problema serio – it is / can be a serious problem

Estar un problema serio is not idiomatic. You would use estar with problemático, mal, etc., but not in this structure:

La situación está seria / está mal – The situation is serious / bad

Why isn’t it subjunctive: ella explica que una hipoteca grande… pueda ser un problema serio?

The subjunctive in Spanish is often used when there is doubt, wish, emotion, uncertainty, or subjectivity about the action in the subordinate clause.

Here, ella explica que… introduces something that she presents as a fact or a general truth:

Ella explica que una hipoteca grande sin buen salario puede ser un problema serio.
→ She explains (states) that this can be a serious problem.

Because the clause que una hipoteca… puede ser un problema serio is being asserted, not doubted or wished, the indicative (puede ser) is the natural choice.

You would use the subjunctive after verbs like:

dudar que, no creer que, es posible que, etc.

Why is problema masculine (un problema serio) even though it ends in -a?

Problema is one of a group of Greek-origin nouns in Spanish that end in -ma but are masculine. Common examples:

el problema – the problem
el programa – the program
el tema – the topic
el sistema – the system
el clima – the climate

So we say:

el problema, un problema serio
este problema (this problem, masculine)

And the adjective serio must agree in gender and number, so it is serio (masculine singular), not seria.

Could we say un serio problema instead of un problema serio? Is there a difference?

Both are correct, but the position of the adjective changes the nuance:

un problema serio – a serious problem (neutral description; adjective after the noun)
un serio problema – a very serious / grave problem (often more emphatic or literary)

Adjectives before the noun can sound:

More emotional or expressive
More subjective

In everyday speech, un problema serio is more common and neutral.
Un serio problema sounds a bit more formal, dramatic, or emphatic.

Is there anything special about how this is pronounced in Latin American Spanish?

A few points for Latin American Spanish:

Ella: usually pronounced like e-ya or e-ja (the ll sounds like the English y in yes in most regions).
Stress:
- E-lla ex-PLI-ca que u-na hi-po-TE-ca GRAN-de sin BUEN sa-LA-rio PUE-de SER un pro-BLE-ma SE-rio.
- The stressed syllables are in caps above.
Que: always a hard k sound at the start: keh, not kwee or kjuː.
Hipoteca: initial h is silent: i-po-TE-ca.

There are regional accents, but the pronunciation described above works across Latin America.