When something bad happens like someone dying or things being stolen, the police try to figure out what happened. Sometimes it's not clear exactly what happened. In this case, the police first guess what kind of crime it is.

For example, if a man died at home alone, there might be a few different explanations. If the police think that the man died from doing drugs, you can say this:

The police suspect a drug overdose.

If they think that someone killed the man, you can talk about it like this:

They suspect murder.