You know, being able to make a microprocessor from scratch and being able to supply the right instructions to get what you want -- these are different domains. As long as you can understanding what Machine Learning is _doing_ I think getting practice with some libraries and watching a lecture or two will give you the ability to play with and train them.
Basically, you have a training set of data that has "this is the data in" and "these are the outputs I expect." so if you are training something to learn the different animals of the animal kingdom, your data set might be portraits of giraffes and gorillas, and your outputs might be simple labels like "gorilla, snake, giraffe, tucan" etc. The neural net part of it is like the parameters you can hone. You can hone how many layers deep it is (try a layer for each dimension, like if you are tracking 5 facial features, you can try a network of roughly 5 layers). You have an "activation function" [usually sigmoid] because the _neurons_ sum their inputs and only fire on activation threshold (like your brain neurons).
Anyway, don't let the maths hold you back. When you can find a Python library on neural networks and play with it well enough to get results, you have pretty much figured out the puzzle. Not many people have deep intuition on how it works exactly, because the parameters (the "brain" of "neurons") is really an abstract mathematical object that is adjusted regressively over the course of learning [the training data].
Don't be discouraged, there is plenty of growth in software, and unless you are trying to help make a breakthrough in ML or "artifical intelligence" which, in my opinion, is something we have not even come close to touching, you don't really need to know the nitty gritty. The promising modern day results in ML came about because someone wanted to model a brain with a computer and ended up learning how to pattern match.
(Results match Data? Accept : Adjust)