
aussagen
to testify
To give an official, usually oral statement before an authority, especially in court or to the police.
Der Zeuge weigerte sich, ohne seinen Anwalt auszusagen.
The witness refused to testify without his lawyer.
to testify / indicate
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to testify
To give an official, usually oral statement before an authority, especially in court or to the police.
Der Zeuge weigerte sich, ohne seinen Anwalt auszusagen.
The witness refused to testify without his lawyer.

to indicate
To mean, express, or contain something specific as information.
Was soll diese Statistik über die Kundenzufriedenheit aussagen?
What is this statistic supposed to indicate about customer satisfaction?
'Aussagen' (to testify / to indicate) is a separable verb. The prefix 'aus-' goes at the end in a main clause. In subordinate clauses and in the infinitive with 'zu', the verb stays together.
Der Experte sagt gleich im Fernsehen aus.
The expert is about to testify on television.
The meaning often depends on the subject. When a person 'aussagt' (testifies), it usually means 'to give an official statement'. When a thing (e.g., a statistic or a text) 'aussagt' (indicates), it means 'to express or show something'.
Der Angeklagte wollte nicht aussagen.
The defendant did not want to testify.
In the meaning 'to give a statement', 'aussagen' (to testify) is almost always used in an official or legal context. Typical phrases are 'vor Gericht aussagen' (to testify in court), 'bei der Polizei aussagen' (to give a statement to the police), or 'als Zeuge aussagen' (to testify as a witness).
Er musste bei der Polizei zu dem Vorfall aussagen.
He had to testify to the police about the incident.
When you make a statement that harms another person or accuses them, you use the preposition 'gegen' (against) with the accusative.
Der Hauptzeuge hat gegen den Angeklagten ausgesagt.
The main witness testified against the defendant.
to express
Both words refer to communicating something. “Äußern” is especially suitable for a personal opinion, feeling, or wish, while “aussagen” is more likely to refer to factual information or a specific meaning. For your own assessment, you generally use “äußern”; for the information contained in a statement, you use “aussagen”.

Der Bericht sagt aus, dass die Straße dringend repariert werden muss.
The report indicates that the road urgently needs to be repaired.
Die Bürgerin äußerte ihre persönliche Meinung zur geplanten Straße.
The citizen expressed her personal opinion about the planned road.