Answer to first part
The function in the question is called the logistic function. Sometimes it is also called "the" sigmoid function, but some authors use sigmoid to just mean any s-shaped function.
There are a wide variety of activation functions used in practice in neural networks, sigmoid and otherwise. The logistic function is one of the more common ones, because both the logistic function and its derivative are defined for all real inputs, can be written as short expressions in terms of elementary functions, and can be computed efficiently using standard library functions in most programming languages. (This is unlike the step function used as the activation function for a classic perceptron—the derivative of the step function is undefined at the discontinuity.) Another widely-used activation function that has these properties is $\tanh$. There is really no strong reason to prefer one over the other when initially presenting sigmoid neurons. You can pretty much pick any function you learn how to differentiate in a Calculus 1 class and that has a sigmoid shape with asymptotes at $\pm\infty$. 915
They have slightly different performance characteristics in training, but that is not very relevant for an initial explanation.
It is unfortunately very difficult to explain backpropagation without understanding partial derivatives, since backpropagation is literally just gradient descent where the gradient is computed by automatic differentiation. I would recommend watching 3Blue1Brown's excellent series of YouTube videos on how backpropagation works: part 1, part 2, and especially part 3 and part 4.
You mention an audience in the question. If you are going to be presenting this material, I would seriously consider referring your audience to the 4 videos linked above, at least as references. They certainly provide a better explanation than I could give in a lecture.
Answer to second part
The reason for not using a linear function is that a neural network with all linear activation functions is a linear combination of linear functions, and is therefore itself a linear function. So using a linear activation function misses the entire point of training a neural network; you could get the same result faster by doing a least-squares fit of a linear function to the data.
To oversimplify only slightly: a neural network with a linear activation function is just the "fit trendline" feature in Excel.
By contrast, there is a universal approximation theorem that says that, for sufficiently nice nonlinear activation functions, any function can be approximated well by using enough neurons.
The universal approximation theorem was not discovered for many years after when neural networks were first invented, so it was not a motivating factor in their invention. Early neural network research was mainly inspired by biological neurons (the kind in your brain) and control theory.
While the universal approximation theorem says that a sufficently large neural network has the potential to approximate any function well, the actual reason why the standard method of training neural networks (stochastic gradient descent backpropagation) performs so well in practice is still poorly understood and an active area of research. 716