That is an interesting video to help understand how the orbits work.
I think they used the boost from how the orbits worked with earth gravity to help sling them to the moon and I heard Jeremy Hansen talk about how it looked like they were racing toward earth when they first headed off to the moon.

Something that bothered me about news stories was the way they worded them 'that they were going to go around the moon and be the first to see the other side of the moon'
They failed to point out that man had gone around the moon several times before and what was different this time was the orbit around the moon allowed them to see areas that had not been seen on the other orbits.
I thought the news stories left the impression that this was the first time men had orbited the moon.
Several news reports in Canada were worded that way not just one outlet .