You need a certain requisite amount of abstract thinking skill before you can make (valid) inferences. According to psychologist Jean Piaget, the ability to engage in abstract thought kicks in from late childhood to early adolescence, when the child is capable of "hypothetical and deductive reasoning." Prior to that, language and abstraction skills are not developed to the point of being able to make valid inferences. Thus, while it is possible to teach inference-making to teenagers, it is not often possible to teach it to very young children (i.e. children younger than 6).
There are many different ways in which a person can draw an inference. Inferences can be drawn from text, from experience or from independent thought. However, all inferences can be placed into one of two broad categories: propositional and pragmatic. Propositional inferences are derived from language (text or spoken word) while pragmatic inferences are derived from experience. Teaching children to make propositional inferences requires, as a prerequisite, that the child has a firm grasp of language. Teaching children to make pragmatic inferences requires that the child have the ability to notice things about the world around them (which can be taught, but not quickly).
Inference is a tool of logical reasoning. When you make an inference, you are essentially deriving a true statement from other true statements. Thus, there are two parts to teaching inference to young children: 1, teaching children how to capture facts in clear sentences, and 2, teaching children how sentences can follow from other sentences. For the first objective, show students to write out all the essential facts in any text they read. For the second, get students to look over all the information they have laid out on paper, then explain how two pieces of information can imply another, unstated piece of information. Use simple examples ("Mary only had blond hair when she was 13," "Mary had blond hair in the story," "therefore, Mary was 13 in the story") to make the rules of inference as clear as possible.
Students will not learn solid inference skills just from having you explain the rules of inference to them. They will need to practice making their own inferences before they can be said to have learned the skill. One activity is to have students read a story and play the role of a detective, writing down each relevant piece of information in the story and using these pieces of information as clues to find out what's written between the lines. In another activity, you have students write a one-paragraph story. However, you make it clear to them they cannot say the main point of the story directly; they have to give clues to the main point by writing two separate sentences from which the main point can be inferred.