La doctora me explicó que debería descansar más y controlar la ansiedad.

Because the sentence is referring to a female doctor. In Spanish, articles and many profession words change for gender:

• el doctor = the male doctor
• la doctora = the female doctor

In Spain, you may also hear la médica for a female physician, but la doctora is very common too.

Me means to me.

The verb explicar often works like this:

• explicar algo a alguien = to explain something to someone

So:

• La doctora me explicó... = The doctor explained to me...

Spanish usually uses an indirect object pronoun like me, te, le, nos, os, les before the verb.

Explicó is the preterite, which is used for a completed action in the past.

So me explicó presents the explanation as a single completed event:

• The doctor explained to me...

If you said explicaba, that would sound more like:

• she was explaining
• she used to explain
• she kept explaining

Here, the sentence is talking about one specific explanation, so explicó fits better.

The accent mark shows the stress and also helps distinguish forms.

• explico = I explain
• explicó = she/he explained

So the accent is important both for pronunciation and meaning.

Que means that here.

Spanish usually keeps que when introducing a clause after verbs like decir, explicar, creer, pensar:

• Me explicó que... = She explained that...

In English, that is often optional:

• She explained (that) I should...

In Spanish, que is normally required.

Debería means should and is the most natural choice for advice or recommendation.

• debo = I must / I have to
• debía = I was supposed to / I should have / I had to, depending on context
• debería = I should

So debería descansar más means:

• I should rest more

It sounds like medical advice, not a strict order.

Because explicar que normally introduces reported information, not doubt, emotion, or a command.

So Spanish uses the indicative/normal finite form here:

• me explicó que debería...

Compare that with a verb of recommendation:

• Me recomendó que descansara más.

After recomendar que, the subjunctive is natural:

• descansara

But after explicó que, the subjunctive is not expected in this sentence.

Because after debería, Spanish uses an infinitive.

Pattern:

• debería + infinitive

So:

• debería descansar
• debería controlar

This is just like English:

• should rest
• should control

You do not conjugate the second verb after debería.

Because más modifies the verb descansar and means more.

• descansar más = to rest more

This word order is very natural in Spanish. It is similar to English:

• rest more

Also, más has an accent mark. That distinguishes it from mas, an old-fashioned/literary word meaning but.

Spanish often uses the definite article with abstract nouns, illnesses, and conditions more often than English does.

So:

• controlar la ansiedad = to control anxiety

Even though English often says just anxiety, Spanish commonly says la ansiedad.

This is very normal with things like:

• el estrés
• la fiebre
• la presión
• la ansiedad

It could, but only if the meaning is already clear from context.

In this sentence, la ansiedad is stated explicitly:

• ...y controlar la ansiedad

That is clearer, especially if this is the first time it is mentioned.

If the anxiety had already been mentioned, you could say:

• La doctora me explicó que debería descansar más y controlarla.

Here, la would refer back to la ansiedad.

In modern standard Spanish, especially in careful usage, the usual recommendation form is:

• deber + infinitive for obligation/advice
  
  • debería descansar

Traditionally, deber de + infinitive is more associated with probability:

• Debe de estar cansada. = She is probably tired.

In real-life Spanish, people sometimes mix the two patterns, but in this sentence debería descansar is the best and most standard choice.

They are two coordinated actions linked by y:

• descansar más
• controlar la ansiedad

Both depend on debería:

• debería descansar más
• (debería) controlar la ansiedad

Spanish often avoids repeating the auxiliary/modal verb when it applies to both parts. So the full idea is:

• She explained that I should rest more and should control anxiety.

But Spanish naturally says it only once.