What is "volume" in $n$ dimensions? It is a certain quantity obtained by calculating a certain measure defined on $\mathbb{R}^{n}$. We know exactly which measure it is - it is the product measure which gives the box $\prod_{i=1}^{n}[a_i,b_i]$ the measure $\prod_{i=1}^{n}(b_i-a_i)$. The fact that we use this measure and not any other measure (there are lots of possibilities) as the "natural" measure on $\mathbb{R}^n$ is probably because
A. If is very easy to calculate this measure of boxes.
B. Many important and interesting subsets of $\mathbb{R}^n$ can be approximated very well by boxes.
We call this measure of sets the volume of sets, although a more proper name should be the content of sets, because volume, really, is a three-dimensional notion. Once we agree on this measure, we find that when we come to measure the volume of the unit-ball, i.e., the set $\{x\in\mathbb{R}^n:\sum_ix_i^2\leq 1\}$
it turns out that the largest copy of the unit-box (i.e. the set $\{x\in\mathbb{R}^n:\max_i|x_i|\leq 1\}$) that fits into the unit-ball has to be dilated by a factor of $n^{1/2}$ (i.e., the unit cube has to be multiplied by $n^{-1/2}$ in order to fit into the unit ball). Now this does not prove anything yet, but it clearly gives some recognition of the fact that the unit-ball in $\mathbb{R}^n$ does not occupy much volume, and that it is much "harder" to be inside a $100$-dimensional ball than inside a $3$-dimensional ball because there are $97$ more inequalities that have to be satisfied. Of course, all this is just wave-handing heuristics, but when you perform the calculation you find a number that goes to zero approximately like $n^{-1/2}$, validating some of the feeling of small volume we got by comparing to the cube - the basic building block of the underlying measure.
As for the question why is there a maximum - there is really a trivial answer: the volume starts somewhere, and then it decreases to zero, so it must have a maximum.
Regarding the third question - why does the surface area attain a maximum in a different dimension than the volume, I don't see any "naturally" good answer. Just as well, I could ask "why should it?"