Jeanene, I have tried about every king of wood filler and not found one that takes stain like the surrounding wood. They all advertise that they are stainable, and they are to some extent, but then you are still faced with using stain markers, tone sprays and the like to get the wood to match.
Burn-in sticks are shellac that you use with an iron that is much like a big soldering iron or wood burning iron with a special tip. You use the iron to heat the shellac so it flows into the defect. Since the sticks come in many different wood tone colors, you can start with the background color of the wood and use the stain markers and tone sprays to get the repair to blend into the rest of the wood.
As you might begin to appreciate, there is a lot of art to this process. It takes a bit of experience and practice to hide repairs reasonably well. It also takes some special products. Since you can easily spend as much for these as for a repair, it usually is close to a wash as far as cost to have something repaired by a pro.
For what it's worth, no, I am not a pro but have done some of this and know some of the tricks of the trade. I also know that a good pro is a heck of a lot better at this than I am.