I tried to do something like the first picture in this article and it was a disaster. I was working with 3/4" maple and bit dug in, chewed it to pieces, and eventually broke the piece in half. I have a fresh, sharp bit, a powerful router, and I was moving the right way past the bit. What did I do wrong? Should the router be on high speed or low? What else affects my chance of success?
It looked something like this: 
This is a very dangerous task if you don't know what you are doing. There are a couple of issues here that can lead to blowout.
First, as Graphus mentioned, is the amount of material you are removing. I would recommend being as close to 1/16" as you can comfortably get with your rough cut.
Second, when you are climbing the curve, so to speak you are going against the grain, and particularly with a straight cutter like is shown in that image, you will be fighting that the whole way. It will want to grab and as soon as it does, its going to pop on you. I prefer to avoid straight cutters in this scenario. I would recommend a spiral cutting bit, to help minimize that blowout and tendency of the bit to grab the material. Freud Spiral Flush Trim
Lastly, for tighter radius parts, it is often worth it to work the piece in halves, flipping the part for each half to always be working with the grain. You can either flip the pattern to the other face, or switch between a flush trim and pattern bits as needed to accommodate.
