A loop $\alpha: I \rightarrow X$ is nulhomotopic $\iff$ $\alpha$ extends to a map $\alpha': D^2 \rightarrow X$

Question

I'm looking to build the intuition as to why this is true. So, i'm viewing the map from $I$ to $X$ as a map from $S^1$ to $X$, and it sort of makes sense that in a simply connect space like $D^2$ every path is nulhomotopic, but still, I feel like something is missing from my understand of this. My professor did this proof without using the fundamental group, but he did go ham into euclidean geometry. Can anyone offer some insight into this for me? Thanks!