Learn Dutch: Ik heb een lucifer nodig om het vuur mee aan te steken.

This is the core of the current lesson. When a preposition refers back to a noun (een lucifer) and gets separated or 'stranded' before the verb, met always changes to mee. It translates exactly to the English 'a match to light the fire with'.

Where does the object het vuur belong in this type of clause?

It goes inside the clause, sandwiched between om and the verb cluster. The standard structure is: om + object (het vuur) + stranded preposition (mee) + te-infinitive (aan te steken).

How is the verb aansteken (to light) formed with te here?

Because aansteken is a separable verb, the te must be placed between the separable prefix (aan) and the main verb (steken). In Dutch, this is written as three separate words: aan te steken.

Why do we use heb ... nodig instead of a verb like moeten?

nodig hebben specifically means 'to need' when you require a noun (like een lucifer). You use moeten when you need or have to do an action, such as actually lighting the fire.