A more thorough explanation:

An assertory oath is a solemn declaration made by a person, accompanied by a swearing to God or a revered person or thing, that their statement is true or that they will be bound to a promise. The person making the oath implicitly invites punishment if the statement is untrue or the promise is broken. The legal effect of an oath is to subject the person to penalties for perjury if the testimony is false.

For example, a witness in a courtroom typically takes an assertory oath to attest to some factual matter. This type of oath is different from a promissory oath, which is a promise about one's future conduct.