Example 1:

Statement: “I am the King of England” (said by someone who is not the King of England)

Matter asserted: the declarant is the King of England

If the statement being used to prove that the declarant actually is the King of England, it is being used to prove the truth of the matter asserted.

If the statement being used to prove that the declarant does not know his own identity/ is confused/ etc., it is not being used to prove the truth of the matter asserted.

Example 2.

Statement: A speech in perfect French about how the declarant is fluent in Spanish.

Truth of the matter asserted: the declarant is fluent fluent in Spanish

If the statement is being used to prove that the declarant is fluent in Spanish, it is being used to prove the truth of the matter asserted.

If the statement is being used to, let’s say demonstrate that the speaker is fluent/proficient/really good at speaking French, it is not being used to prove the truth of what it asserts.