When you use a separable verb, you usually have to take off the prefix and put it at the end of the clause.
anziehen - to put on
fernsehen - to watch TV
In an um...zu... clause, the infinitive has to be at the end. But when using a separable verb, you 'separate' the prefix and the rest of the infinitive by inserting zu between them, so that they sit together as one word at the end of the sentence.
The following examples will explain it more clearly, where you can see that the zu is embedded within the separable verb.
Ich gehe ins Wohnzimmer, um eine Stunde fernzusehen. – I go into the lounge in order to watch TV for an hour.
The zu has forced its way between the prefix and the rest of the infinitive and snuggled in nicely.
Link these sentences with um...zu… in the middle, then translate them into English to check your understanding.
Link these sentences with um...zu... at the beginning.