This is a harder problem than it sounds, and something I've given a lot of thought to. I think the underlying issue is that all machine learning was discovered through a combination of applied math and intuitive ideas/models. Without the intuitive model no one would have thought to discover the method, and without the math the intuitive idea would be a pipe dream. Both are fundamentally linked, and it's a bad idea to separate them.
For example, one of the most basic and oldest statistical methods is linear regression. The first thing anyone will tell you when they are teaching it is the basic "idea and model" - finding a line that fits a scatter plot (and then extending that idea). But this doesn't give you a real understanding of linear regression: where does that line come from, and why does a clean, algorithmic solution exist? Why does the standard solution often lead to numerical errors, and why is regularization a valid solution?
These questions require an increasing amount of math, but they are essential to really understanding linear regression. I agree that some textbooks just throw you a wall of math, but many actually do a solid job of explaining what is going on as they do so.
If you don't want to know the details of machine learning methods, which are inherently mathematical, you might as well as just remember the names of libraries that implement the solutions for you.