The good news is that your instincts are right that it would be a useful technique. The bad news is that it's not a technique that you can use without understanding a fair amount of linear algebra. It's all about multiple equations with multiple matrix multiplications.
Some tools like R's bsts package make it more accessible, but it's fundamentally more complex than alternatives. Not that you should be using ARIMA or other methods without some level of technical sophistication, but in my experience most state space (also called dynamic linear model) packages have gaps where you'll need to know what parts of various matrices represent and mean.
Given all of that, as a readable introduction I'd recommend "An Introduction to State Space Time Series Analysis" by Jacques J.F. Commandeur and Siem Jan Koopman, Oxford 2007. It's a short book and used to be pretty expensive, but it appears that it may have been released on the Internet. I don't believe this book mentions mixed-frequency data, though.
And if you use R, you should check out bsts.
