Learn Hardcore German: Die Trainerin nutzt die Halbzeit, um den Kern ihres Plans zu erklären. - The coach female uses half-time in order to explain the core of her plan.

Die Trainerin nutzt die Halbzeit... (natural, focuses on using the time/opportunity)
Die Trainerin benutzt die Halbzeit... (grammatically okay, but sounds a bit off; you usually don’t benutzen a period of time)

So nutzen is the natural verb here because halftime is an opportunity, not a tool.

Why is it die Halbzeit and not der Halbzeit or dem Halbzeit?

Halbzeit is a feminine noun: die Halbzeit (halftime).

Case and role in the sentence:

Verb: nutzt (uses)
Direct object of nutzt: die Halbzeit

The direct object is in the accusative case. For feminine nouns, the article is:

Nominative: die Halbzeit
Accusative: die Halbzeit

So you see die in both nominative and accusative for feminine nouns.
It is not dative (der Halbzeit) here, because nutzen takes a direct (accusative) object, not a dative one.

What is the function of um ... zu in um den Kern ihres Plans zu erklären?

um ... zu introduces a purpose clause: it expresses in order to / so as to.

Structure:

um
- accusative object(s)
  - zu
    - infinitive (verb)

In the sentence:

um den Kern ihres Plans zu erklären
= in order to explain the core of her plan

So the whole sentence is:

She uses the halftime in order to explain the core of her plan.

Without um ... zu, the sentence would lose that clear purpose meaning.

Why is there a comma before um?

In German, um ... zu introduces a type of subordinate clause (a non-finite clause expressing purpose).

Rule: Subordinate clauses are separated from the main clause by a comma.

Here:

Main clause: Die Trainerin nutzt die Halbzeit
Subordinate clause (purpose): um den Kern ihres Plans zu erklären

So you must put a comma:

Die Trainerin nutzt die Halbzeit, um den Kern ihres Plans zu erklären.

Why is it den Kern and not der Kern?

Kern (core) is a masculine noun:

Nominative: der Kern
Accusative: den Kern
Dative: dem Kern
Genitive: des Kerns

In the um ... zu clause, erklären (to explain) takes a direct object:

etwas erklären = to explain something

That direct object is den Kern, so it needs the accusative:

den Kern = the core (as direct object)

So:

Subject (implied): sie (the trainer)
Verb: zu erklären
Direct object: den Kern (accusative)

What exactly does Kern mean here?

Literally, Kern means core, nucleus, or kernel (like the core of an apple or the nucleus of an atom).

Metaphorically, it is often used as:

Kern eines Problems = the core of a problem
Kernaussage = main statement / central message

In this sentence, den Kern ihres Plans means the central idea or the essential part of her plan, not every tiny detail.

How does ihres Plans work grammatically, and why not ihrer Plan?

Plan is masculine: der Plan.

In den Kern ihres Plans, Kern is the noun in the accusative:

den Kern = the core

What kind of core? The core of her plan → this is a genitive relationship:

der Kern des Plans = the core of the plan
With a possessive: der Kern ihres Plans = the core of her plan

Forms:

Masculine singular genitive with possessive ihr-:
- des Plans → ihres Plans

So:

ihr = base form (her / their / your (formal) depending on context)
Genitive masculine singular: ihres
Noun: Plans (Plan + -s in genitive)

ihrer Plan would be wrong here because:

ihrer is genitive feminine or dative plural form of ihr-
Plan is masculine, so it needs ihres Plans in the genitive.

Does ihr here mean her, their, or your (formal)?

In isolation, ihr (and its forms like ihres) can mean:

her (belonging to a female person)
their (belonging to several people)
your (formal, belonging to Sie)

In this sentence, context decides:

The subject is Die Trainerin (the female coach)
So ihres Plans naturally refers back to her

Therefore, here ihres Plans means of her plan, not of their plan or of your plan.

Why is the verb erklären at the very end: zu erklären?

In um ... zu infinitive clauses, the infinitive verb goes to the end of the clause.

Pattern:

um
- (objects etc.) + zu
  - infinitive

Examples:

Er geht zur Bibliothek, um zu lernen.
Sie ruft ihn an, um eine Frage zu stellen.

In this sentence:

um den Kern ihres Plans zu erklären
- Objects: den Kern ihres Plans
- zu
  - infinitive: zu erklären at the end

So zu erklären must appear at the end of the um-clause.

Could we express the same idea without um ... zu?

Yes, a very common alternative is a damit clause, which also expresses purpose:

Die Trainerin nutzt die Halbzeit, damit sie den Kern ihres Plans erklärt.
(The trainer uses halftime so that she explains the core of her plan.)

Difference in feel:

um ... zu: more compact, no explicit subject, often a bit more neutral/formal.
damit: has its own conjugated verb and subject, often used when you want or need to show who is performing the action in the purpose clause, especially if it’s a different person.

But for this specific sentence, um den Kern ihres Plans zu erklären is very natural and idiomatic.

Why are Trainerin, Halbzeit, Kern, and Plans capitalized?

In German, all nouns are capitalized, regardless of position in the sentence.

In this sentence:

Trainerin – noun (female coach)
Halbzeit – noun (halftime)
Kern – noun (core)
Plans – noun in genitive (of plan)

Verbs (nutzt, erklären) and other parts of speech are written in lowercase (unless they are at the beginning of a sentence or are part of a proper name).