What is the interval of definition of a solution of an ODE? I don't understand this question.
Verify that the indicated function $y=\phi(x)$ is an explicit solution of the given ODE. Consider $\phi$ simply as a function, to give its domain. Then by considering $\phi$ as a solution of the ODE, give at least one interval $I$ of definition.
The answer key gives the largest domain of function as  $[2, \infty)$ and largest interval  of definition for solution is $(2, \infty)$
What does interval of definition mean?

An interval of definition of a solution is any (open) interval on which it is defined. For example: the problem $y'=y^2$, $y(0)=1$, has solution $y(t)=1/(1-t)$. This solution is defined on the interval $(-1/2,1/3)$, so we can say this is an interval of definition of solution. It's not the largest such: the largest interval of definition is $(-\infty,1)$. 
Another example: $y(t)=\sqrt{t}$ satisfies the ODE $y'=1/(2y)$. Although the domain of square root is $[0,\infty)$, when it is considered as a solution of this ODE, the interval of definition is $(0,\infty)$. At $t=0$ the ODE does not make sense, so it would not be correct to say that $y$ satisfies the ODE there. And generally,  ODEs are considered on open intervals only. 
Also, I'd expect the book you are using to define the term interval of definition the first time it appears.
