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YouTube's help section tells you very little about how to solve it.
Thankfully, it's pretty simple.
Assuming you've been given some HTML to embed into your website, such as:
<iframe src="https://www.youtube.com/embed/?list=UULFZXXi0dM4r7-sKFNo1kGlwg" width="480" height="400"></iframe>
Then you only need to add one attribute,
allowfullscreen
:<iframe allowfullscreen src="https://www.youtube.com/embed/?list=UULFZXXi0dM4r7-sKFNo1kGlwg" width="480" height="400"></iframe>
And that's it! It works!