1. It's not "free work", it's just ... stuff you do. Maybe you release it out and other people benefit from it, maybe you just do it to demonstrate skills.
2. Per #1, you get to demonstrate whatever you want to learn. Do you want to learn how to use Twilio APIs? You could wait for a client engagement that might need Twilio, or you could try to force Twilio in to a client project where it's not needed, or you could... just play with Twilio, produce something demonstrable, and use that to show what you know about using Twilio. It's great when you can show your skills having real value to real past client projects, but absent that demonstration and experience, you'll be experimenting on your new clients to a degree. Learning on your own dime (and time) can also inspire a degree of confidence that you actually know XYZ before you're engaged to do XYZ.
The underlying psychology shows us that we undervalue ourselves whereas other people tend not to since they view us as experts. They are generally willing to pay a lot more to have a solved problem than we would want to charge.
Start high and go higher until you meet resistance.
This does not seem to mesh with the posted article where they interviewed lots of successful people who started out charging on the low side and were able to aggressively increase over time.
I get what you're saying in this thread, that people tend to not charge enough. Sure, I'm right there with you. What bugs me is this meme that seems to suggest that there's something wrong with doing whatever you can (like charging less) to get your foot in the door. It's intimidating.
Many times what would happen is someone would get two or three bids and avoid the lowballer because quality is associated with higher pricing. So you're shooting yourself in the foot.
Now this doesn't always happen but I would suggest to stick on the side of trying to charge more than you're comfortable with. You'll be surprised what you can get your foot in the door with.
But you need a history of engagements to build a client portfolio.
