I know that,
$\sum_{i=0}^{n} i = \frac{(n)(n+1)}{2}$
because this is a summation property. But can someone please explain to me how this is found. I am confused about the two different variable start and end conditions. I have not seen any similar questions asked on math.stackexchange and when I try to google it there are no results. Please help.
In the series, $\sum_{i=0}^n i$ only the variable $i$ is the iteration variable, while the variable $n$ is the boundary condition. You can treat the later as a constant for the purposes of the summation.
$$\begin{align}\sum_{i=0}^n i ~&=~ 0+ 1+\cdots + (n-1) + n \\[1ex] &=~ \begin{cases}(0+n)+(1+n-1) +\cdots +(\tfrac{n+1}2+n-\tfrac{n+1}2) &:& n\text{ is odd} \\ (1+n)+(2+n-1)+\cdots+(\tfrac n2+n+1-\tfrac n2)&:&n\text{ is even} \end{cases} \\[1ex] &=~ \begin{cases} \sum_{j=1}^{(n+1)/2} n &:& n\text{ is odd} \\ \sum_{j=1}^{n/2}(1+n)&:&n\text{ is even} \end{cases} \\[1ex] &=~ n(n+1)/2 \end{align}$$
One way to think about this summation is to find the average of all those numbers, which is $\frac{n+1}{2}$ since the average of $1$ and $n$ is $\frac{n+1}{2}$, the average of $2$ and $n-1$ is $\frac{n+1}{2}$, and so on (perhaps only averaging one item instead of $2$ for the last term if $n$ is even). Then multiply the average by the number of things in your series $n$.
Once a person has an idea for a summation formula, they can always verify its veracity by induction, though I don't think that is quite what you were asking.