Explain why a gamma random variable with parameters $(t, \lambda)$ has an approximately normal distribution when $t$ is large. Explain why a gamma random variable with parameters $(t, \lambda)$ has an approximately normal distribution when $t$ is large.
What I have come up with so far is: 
Let $X=$ the sum of all  $X_i$, where $X_1 = X_1, X_2, \ldots , X_i \sim \mathrm{Exp}(\lambda)$. Then because of the CLT as $t$ gets sufficiently large, it can be approximated as a normal distribution.
Is this sufficient or am I missing a key piece of this proof? 

That's pretty much it except that you assumed $t$ is an integer and you used "$i$" rather then "$t$" at one point.  One way of dealing with non-integer values of $t$ is to go back to the proof of the CLT that uses characteristic functions and make a minor modification in the argument to accomodate non-integers.
(BTW, one writes $Y\sim\mathrm{Exp}$, with that entire expression in MathJax and with \mathrm{} or the like, not $Y$ ~ $Exp$.)
