Flatness of schemes

I am learning about flatness for the first time and I cannot wrap my head around why the definition with tensor products of a flat module implies geometrically that 1-parameter families of schemes have limits.

I came across these lecture notes by Alexander Ritter which give a very explicit example:

However, I don't understand why $\pi$ flat over $0 \iff$$X_0$ is $\lim_b X_b$. Any help to understand this particular example or a different intuition of this geometric implication would be very much appreciated.