I had the same problem you faced a number of years ago... I found lots and lots of resources online about the finite element method that were way too mathematically advanced for me at that time. What I eventually discovered is the following:
There are really two ways to approach the study of the finite element method: the engineering approach and the mathematical approach. Both are valid and arrive at the same conclusion, but have very different starting points. The mathematical approach requires a lot more background in functional analysis, measure theory, variational calculus, etc... It can be very daunting to learn it from a mathematical perspective if you don't already have this background. The engineering perspective uses principles such as virtual work and superposition, and 'shape functions', concepts that are probably more intuitive for a computer scientist's perspective such as yourself. Particularly, the so-called "Direct Stiffness Method" is the simplest approach for beginners in engineering. While the examples tend to be skewed towards mechanical engineering applications such as structural deformation problems, the direct stiffness method can also be used to study problems such as resistor network problems (which are somewhat related to computer science via electrical engineering). It also provides insight into a more generalized approach using the weak formulation and galerkin projection.
Most of the FEM books that I've read with an engineering emphasis tend to focus on HOW it works, and place very little focus on WHY it works. To gain some insight into why it works, you will need some mathematical tools from functional analysis. While it's now out of print, I found the book "Applied Functional Analysis and Variational Methods in Engineering" by Reddy to be particularly insightful and enlightening because it filled in all the "mathematical gaps" from an engineer's persective. Also, Mark Gockenbach's "Understanding and Implementing the Finite Element Method" was particularly helpful to me to understand the convergence theory behind the method and provides nice outline into data structures and matlab code to implement the method.
While I don't know of any particularly useful online resources, I hope I've at least given you some perspective into how to find good resources from a beginner's perspective.