The theory of direct and indirect objects can basically be applied to the German language. See this question for reference. However this is not important for this question.
A subject or an object is usually a noun or a pronoun. But there are also cases in which a subordinate clause is used instead. This clauses are called "Subjektsätze" and "Objektsätze". We are interested in the latter. Here are some examples (objects are marked):
Sie weiß nicht, was ihr Vater gefunden hat.
Er ist sich (dessen) bewusst, dass Probleme auf ihn zukommen werden.
Ich helfe, wem ich will.
Sie ärgert sich (darüber), dass sie den Schlüssel zu Hause vergessen hat.
Now you will probably have to ask:
How can I recognize object clauses?
Well, they replace an object, so you can ask for them with the appropriate question. There are four types of objects:
"Wen/Was?" is the question to ask. Hence, for the first example you would ask: "Was weiß sie nicht?". And the answer would be: "Was ihr Vater gefunden hat."
"Wessen?" is the question to ask. Hence, for the second example you would ask: "Wessen ist er sich bewusst?". And the answer would be: "Dass Probleme auf ihn zukommen werden."
"Wem?" is the question to ask. Hence, for the third example you would ask: "Wem helfe ich?". And the answer would be: "Wem ich will."
For the fourth example you would ask: "Worüber ärgert sie sich?". The answer would be "Dass sie den Schlüssel zu Hause vergessen hat." A prepositional object is connected to the verb via a preposition which can sometimes be omitted. You ask after it by turning the preposition into a question (e. g. "darüber" → "worüber?").
To conclude, you can recognize object clauses by asking the question "Wen?/Was?/Wessen?/Wem?" or turning a preposition into a question. Moreover, object clauses are usually introduced by a conjuction (like "dass" or "ob") or a question pronoun (like "was", "wem", "welcher") and comma-separated (however there are exceptions, too).