qu'ils means 'that they'. qu + apostrophe stands for 'que', ils is the third person plural.

je sais qu'ils sont ici> I know that they are here

"puissent" is a form of the verb "pouvoir", to be able to, can.

It is the third person of plural in the present of the subjonctive, "qu'ils puissent."

It is used to express a wish:

Je souhaite qu'ils puissent rentrer, I wish they are able to come back.

A request:

Il faut qu'ils puissent rentrer, they have to be able to come back.

A possibility:

Il est possible qu'ils puissent rentrer, they may be able to come back.

A doubt:

Je doute qu'ils puissent rentrer, I doubt they will be able to come back.