Learn Swedish: Hon ser honom närma sig och hör honom vissla. - She sees him approach and hears him whistle.

This sentence uses verbs of perception (se and höra). Just like in English ("sees him approach"), when you see or hear someone doing something, Swedish uses the object (honom) followed directly by the dictionary form of the action verb (the infinitive, like närma and vissla).

Why isn't there an att before närma and vissla?

Verbs of perception like se (to see) and höra (to hear) take a bare infinitive. This means you do not use the infinitive marker att (to). It works exactly like English: you say "hears him whistle," not "hears him to whistle."

Why is the word sig used here, and who does it refer to?

The verb att närma sig (to approach) is reflexive, so it always needs a reflexive pronoun. Even though Hon is the main subject of the sentence, the action of approaching is being done by the object, honom (him). Since "he" is the one approaching, the reflexive pronoun must match him, which is sig.

In English I could say "She sees him approaching." Can I use an -ande word in Swedish here?