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A topological space X is said to be simply connected if any loop $\alpha(t)$ (defined as a map $\alpha : [0,1]\to X$ and $\alpha(0)=\alpha(1)=x_0$) in X can be continuously shrunk to a point. What is the mathematical meaning of shrinking a loop to a point in homotopy theory?

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    This is used as a way of classifying spaces. Sometimes on spaces there are inherently curves that cannot be shrunk to a point (a torus for instance). This is because the torus naturally has *holes* in it (the torus in a sense is wrapped around space in a nontrivial way). A space is simply connected if all the loops can be shrunk to a point, or in a sense there are no holes in the space. – 2017-01-26
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    More generally, if $S\subset D$ are two spaces, we can ask when a continous function $f:S\to X$ can be extended to a continuous function $g:D\to X$. In this case of a loop, which can be seen as a map $S^1\to X$, we might ask this question with $D$ equal to unit disk containing $S^1$ as the boundary. So in some sense, the prototypical loop shrinking to a point is the circle shrinking towards the origin. – 2017-01-26

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This means that you can make $\alpha$ a member of a family of curves which include the constant curve as a member.

Think of $\beta_0(t)=\alpha(t)$. We want to deform $\beta_0=\alpha$ until it is $\beta_1(t)=x_0$ (the constant boring curve). In other words, we want a continuous function: $\beta:[0,1] \times [0,1] \to X$ such that $\beta_0(t)=\beta(0,t)=\alpha(t)$ ($\beta$ "starts" off as $\alpha$). Next, $\beta_1(t)=\beta(1,t)=x_0$ ($\beta$ "ends" up being the constant curve). Finally, we want all of the intermediate curves to start and end in the same way as both $\alpha$ and the constant curve -- that is -- we want $\beta_s(0)=\beta(s,0)=x_0$ and $\beta_s(1)=\beta(s,1)=x_0$ for all $s \in [0,1]$.

We imagine the curve $\beta_0=\alpha$ being continuously deformed into $\beta_1$ (the constant curve) while always requiring the beginning and endings of the intermediate curves being fixed at $x_0$.

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The precise definition is as follows. A homotopy of loops at $x_0$ in $X$ is a continuous map $H:[0,1]\times[0,1]\to X$ such that $H(0,t)=H(1,t)=x_0$ for all $t\in [0,1]$. Intuitively, we think of $H$ as consisting of a loop $H_s(t)=H(s,t)$ for each $s\in [0,1]$ which "varies continuously" with respect to $s$. We then say a loop $\alpha$ "can be shrunk to a point" (or more formally, is nullhomotopic) if there exists a homotopy of loops $H:[0,1]\times[0,1]\to X$ such that $H(s,0)=\alpha(s)$ and $H(s,1)=x_0$ for all $s\in[0,1]$. We think of $H$ as giving a continuous deformation that starts with the loop $\alpha$ and ends with the loop $H_1$ which just sits at $x_0$ and doesn't move.

(Note that the definition of simply connected is not just that every loop can be shrunk to a point: you also need to require that $X$ is path-connected.)