How do layers in an artificial neural network transform inputs to outputs? To me, most ANN/RNN related articles don't tell me actually how the network is implemented. I know that in the ANN you'll have multiple neurons, activation function, weights, etc. But, how do you, actually, in each neuron, convert the input to the output?

Putting activation function aside, is the neuron simply doing $\text{input}*a+b=\text{output}$, and try to find the correct $a$ and $b$? If it's true, then how about where you have two neurons and their output ($c$ and $d$) is pointing to one neuron? Do you first multiply $c$ and $d$ then feed it in as input?
The basic calculation for a single neuron is of the form

$$\sigma\left(\sum_{i} x_i  w_i \right),$$

where $x_i$ is the input to the neuron $w_i$ are the neuron-specific weights for every single input and $\sigma$ is the pre-specified activation function. In your terms, and disregarding the activation function, the calculation would turn out to be

$$c\,a_c + d\,a_d + b$$

Note, that the bias term $b$ is just a weight that gets multiplied by the input $1$, thus it appears to have no input.

If you want to develop a further understanding for this, you should try to get familiar with matrix and vector notations and the basic linear algebra that underlies the feed-forward neural networks. If you do, an entire layer of neurons on a whole batch of data will suddenly simply look like this:

$$\sigma(WX)$$

and a FFNN with say 3 layers will look like this:

$$\sigma_{3}(W_3\sigma_2(W_2\sigma_1(W_1X)))$$