I recently watched a kurzgesagt video on skyhooks, and it mentions that a system of skyhook, one on Mars and one on Earth, can shorten a trip to Mars from 7-9 months to 3-5 months. I am confused on how this is possible. From all I understood, skyhooks simply make it easier to get to orbit, not decrease the time of interplanetary travel. Also, I was wondering that if we had a system of skyhooks orbiting the sun, then maybe we could some how use these skyhooks to possibly bypass launch windows and be able to fly to other planets at any time. The way I thought this would work is that if another planet is on the other side of the sun, then we could maybe slingshot a payload to a skyhook, which would then send it to another, and another, until it reaches its destination. Would this be possible?
This is correct, but not because of Skyhooks being magic.
The most fuel efficient transfers to mars take 7-9 months, but for relatively small increases you can half that, and in some cases do even better. See question and answers here.
A Skyhook can potentially provide a boost at departure and capture at mars, allowing a given craft to use onboard fuel to achieve a higher energy transfer and do the trip in shorter time.
The question is of course if you would, or instead do a slower trip with more payload.
So a Skyhook is a method to provide additional energy to craft, making them 'better', but does not automatically make them faster unless you choose to do so by launching with more fuel than fuel optimal trajectory would require.