Indirect statements are a mystery to me - can you explain them?

First, we need to understand what an indirect statement is, and the best way to do that is to work in English for the moment.

The opposite of an indirect statement is a 'direct statement'. An example might be 'Pompey is a great man'. This is 'direct' because it is simply a statement.

What makes an 'indirect' statement 'indirect', however, is the fact that it is not reported directly, as the above was. For example, I might say the following: 'Caesar said that Pompey was a great man'. The latter is, you could say, 'one step' away from Caesar's proclamation, thus making it indirect.

English nowadays generally expresses this idea as above; x says that y is the case. What English used to do, however, and what you might find in 19th and 20th century literature, is the same idea expressed like this: Caesar said Pompey to be a great man. To my mind, thinking about English in this way is the most helpful way to understand the Latin.

In Latin, indirect speech is characterised by the accusative case + the infinitive, exactly the same way as the alternative way that English used to express an idea. In Latin, the above example would be:

Caesar dixit Pompeium esse magnum virum.

You will see here that 'Pompeium' is in the accusative case, and the verb - here, 'to be' - is in the infinitive. This is the same in the second English example: Caesar said Pompey to be a great man. This also means that anything agreeing with 'Pompeium' - 'magnum' and 'virum' - has to be accusative too.

This means that there's no word in Latin for 'that' as you might find in the English version 'Caesar said that Pompey was a great man'. You should certainly not use the Latin 'ut', as you might think.

Note: there are lots of ways to introduce an indirect statement. In the above example - and a very common one at that - the sentences have a verb of 'speaking'. Here, one 'says' or 'speaks' that something is the case/something to be the case. One can also 'think', 'hear', 'feel', and many others. 

Note also: The tense of the indirect statement is found in the tense of the infinitive. For example, if one thinks that something had happened, a perfect infinitive is used. If one thinks that something is happening, a present infinitive is used. If one thinks that something will happen, a future infinitive is used.

For a list of different types of infinitive, see the second page of this open access explanation: http://www.thelatinlibrary.com/101/IndirectSpeech.pdf

Top Tips:

1) Translating English to Latin

If you read an indirect statement, 'translate' it into the second English version. For example, 'the slave thought that his master had entered the forum' becomes 'the slave thought his master to have entered the forum'. This will remind you to use the accusative and infinitive.

2) Translating Latin into English

Look for the magic combination of accusative and infinitive. This will tell you that you are in indirect speech, and you can translate accordingly. 

Examples:

1) imperator dixit hostes esse molles.

The general said that the enemies were weak
or
The general said the enamies to be weak.

b) Cicero audivit Caesarem appropinquare ad urbem.

Cicero heard that Caesar was approaching the city.
or
Cicero heard Caesar to be approaching the city.

c) coquus putavit dominum mox rediturum esse.

The cook thought that the master would soon return.
or
The cool thought the master to be about to return soon.

- Note the future infinitive here.

These are the basics, but there are further rules to learn, which we can cover in more detail, and practise, in a live session.