What makes food dyes heat-safe?
Nothing makes them heat-safe, that's how they are. 

There are many different molecules which can have a given color. Each of them has its own structure and chemical properties. A food chemist creating a food coloring has to choose his molecules for a ton of different properties, like safety, stability over time, stability in the pH range commonly found in food, heat stability, solubility in water, alcohol and fat, supply, difficulty of processing, cost effectiveness, legal status, non-reactivity with other dyes (to create a mixed color) etc. etc. It is much easier to work if one of those constraints - heat stability - is removed from the equation. 

Wilton's icing colors tend to be used by professionals who want a very wide palette to choose from, have additional requirements on quality, and are specialized enough to use these colors only in icing. This is different from, say, Dr. Oetker customers, who buy a few basic colors and have fun with their kids with them. So Wilton has no need for heat stability for this product, and does not constrain their food chemists to make products which comply to it. 

You cannot make colors heat stable at home, no more than you can keep butter from being liquid at 100 Celsius. It is a basic physical property of the matter you are working with. And yes, most pigments including betalains (in red beets) and anthocyanins are very heat unstable. Carotenoids tend to hold up the best, but I can't think of an example for using lightly processed carotenoid-colored plants (e.g. juiced) to color sweet baked goods. You could try something like making your own annatto if you have a source for the raw fruit and the taste fits your food and you don't mind using a time-consuming process, but it is not clear how this is superior to buying ready made annatto as a condiment, or even just buying the coloring and skipping the "has its own taste" part. 

In the end, making dyes at home is difficult, the results are usually unpredictable and rarely have all qualities people are looking for in dyes. If you want to color your food, commercial dyes work better. If you don't trust those, you are very limited in the results you can achieve.