Short answer: to support prorating and allow a buffer for apparel/gear/merchandise/concessions charges etc
The first reason for authorizing more than your membership cost is to support prorating. If you sign up for a $150/month membership on the 2nd day of the month, your prorated payment will be something like $145. Then on the 1st of the following month you will make a full $150 payment. The two payments, totalling $295, are made within the same "1 month" recurring window. So we authorize 2x the cost of the monthly membership.
In addition we've added $100 to the limit so that, like many new athletes, you want to gear up in that first month (jump rope, t-shirt etc) the gym will be able to charge your account instead of you needing to re-authorize for every charge. You can think of this kind of like a card on file.
So in total, you will be authorizing 2x the membership price + $100. Of course you will only be charged for the actual membership costs and for additional items if you take them.
This is another security precaution that Amazon requires and is the total amount that the gym can ever charge you over the course of the authorization, which happens to be 10 years long.