erklären
to explain
To make a situation understandable to someone or explain how something works.
Der Lehrer erklärt den Schülern die mathematische Formel.
The teacher explains the mathematical formula to the students.

to explain / declare
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to explain
To make a situation understandable to someone or explain how something works.
Der Lehrer erklärt den Schülern die mathematische Formel.
The teacher explains the mathematical formula to the students.

The verb consists of the prefix 'er-' and the verb 'klären' (to clarify/make clear). The prefix 'er-' often expresses the achievement of a goal or result. 'Erklären' (to explain/declare) therefore means creating the result of clarity.
Können Sie diesen Punkt bitte noch einmal klären?
Could you please clarify this point once more?
'Erklären' (to explain/declare) is an inseparable verb. The prefix 'er-' remains firmly attached to the verb stem in all tenses and sentence constructions.
Er erklärte mir geduldig den Weg.
He patiently explained the way to me.
When you explain something to someone, the person is in the dative and the thing is in the accusative. The pattern is: 'jemandem (dative) etwas (accusative) erklären' (to explain something to someone).
Ich erkläre dir die Regeln des Spiels.
I am explaining the rules of the game to you.
This meaning is used in formal and official contexts such as politics, law, or business. The object is often an abstract noun, e.g., 'den Krieg erklären' (to declare war), 'seine Absicht erklären' (to declare one's intention), or 'den Notstand erklären' (to declare a state of emergency).
Das Land erklärte seinem Nachbarn den Krieg.
The country declared war on its neighbor.
In the reflexive form 'sich erklären' (to be explained), the logical puzzle or phenomenon is the subject of the sentence. The reason is often introduced with the preposition 'durch' (by/through).
Sein seltsames Verhalten erklärt sich durch seine Müdigkeit.
His strange behavior is explained by his tiredness.
In this meaning, the preposition 'für' is always used, followed by an adjective or participle. The structure is: 'jemanden/etwas für ... erklären' (to declare someone/something ...).
Der Richter erklärte den Angeklagten für schuldig.
The judge declared the defendant guilty.
to say
'sagen' and 'erklären' can both mean that someone passes on information. 'sagen' is more general and often brief: you say an answer, an opinion, or an instruction. 'erklären' means more: you make something understandable and often give reasons, steps, or details. Learners confuse the words because, when you explain something, you naturally also say something. But not every act of saying something is an explanation. If someone only gives a piece of information, 'sagen' fits. If someone helps someone understand something, 'erklären' is more precise.

Die Lehrerin erklärt der Klasse, wie die Aufgabe funktioniert.
The teacher explains to the class how the task works.
Meine Schwester sagt mir, wann sie am Montag kommt.
My sister tells me when she is coming on Monday.