The definition of a prime ideal is:
An ideal $I$ in a ring $R$ is said to be a prime ideal if $I \neq R$ and $f.g \in I \implies f \in I$ or $g \in I$
My understanding of this is that $f.g \in I$ means atleast $f$ or $g$ has to be in $I$ and we can't have a case where two non-elements of $I$ combine to give us an element of $I$. But surely we can also have both $f,g \in I$ or is it only one element that has to be in $I$ (Sorry if the question seems confusing, but this is the best way I could put it)
You might want to have a look at this page. In particular you might be interested in the truth table $$\begin{array}{|c|c|c|}\hline f&g&f\lor g \\\hline T&T&T \\\hline T&F&T\\\hline F&T&T \\\hline F&F&F\\\hline \end{array}.$$
What this table says to you is that if the statement "$f$" and the statement "$g$" are true then the statment "$f$ or $g$" is true, because both $f$ and $g$ are true.
Similarly if the statement "$f$" is true and the statement "$g$" is not true, then the statement "$f$ or $g$" is true, because $f$ is true.
I let you do the other cases.
In your specific case the statement "$f$ or $g$ must lie in $I$" is the third column of the table. I.e.: it is true if $1)$ both $f\in I$ and $g\in I$ is true, or $2)$ if $ f\in I$ but $g\notin I$, or $3)$ if $f \notin I$ but $g\in I$.
There is also a joke, which might help you understand the concept:
"The wife of a logic has barely had a child. She asks her husband: "is he male or female?" He answers: "Yes"."
Hope it helps.