Before I got into tech, I worked as a full-time tutor with high school and middle school students for a few years. That was the best practice for improving my ability to explain things. I usually think about the meta-game of improving in roughly three different ways:
1. Explaining many things to many people: you really get good at thinking on your feet, reading how well different people understand things, and coming up with new ways of explaining things. It forces you to be flexible, and you can build a general framework for walking through something new.
2. Explaining many things to one person: you learn to see how that person learns. What kind of prior knowledge do they have, and how do they like to work through understanding something new? You can tailor your explanations to them, and since explaining requires a lot of active listening to make sure the person is with you, you can get much better at identifying when they're lost so you can circle back.
3. Explaining the same thing to many people: you hone your explanation skill for that one thing. Once you've explained the concept to multiple people, you can see what works and what doesn't work. You understand the sticking points, the easy parts, the analogies that work, and the ones that don't. Having multiple students working on the same concept at the same time would be great! I'd have to work harder for the first student, but by the end of the week I'd have it down pat.
For me, it came down to practice, repetition, and really knowing the material. Striking the balance between explaining and active listening for ~8 hours a day was a fantastic way to get better, and I don't think I'll ever forget how to walk someone through the tricky parts of AP Calculus. Now that I'm far from the tutoring world, I try to practice by writing and will try to keep that part of my brain active in the future through some sort of teaching.
I'd recommend practicing as much as you can. Keep practicing your explanations and active listening with your daughter. As another user said, make sure you know the material. Walk through it yourself, and try a dry run of explaining some math. I recommend explaining to a pet, which is the equivalent of rubber duck debugging. You won't get the feedback, but you'll be able to think it through first, which absolutely helps. Writing also doesn't offer you a similar feedback loop, but it does force you to walk through your thoughts and identify any missing parts of your own understanding.
Disclaimer: I mostly worked 1:1 with students. Managing an entire classroom is an entirely different beast!