Results of this form, and my intuition from them, come from the Kervaire-Milnor paper on exotic spheres. (There was never a homotopy spheres II. The purported content of that unpublished paper appears to be summarized in these notes, though I haven't read them.) I'm going to need to jump into the algebra here; personally, I couldn't tell you the difference between $S^{57}$ and $S^{61}$ without it.

For $n \not\equiv 2 \bmod 4$, there is an exact sequence $$0 \to \Theta_n^{bp} \to \Theta_n \to \pi_n/J_n \to 0.$$ For $n=4k-2$, instead we have the exact sequence $$0 \to \Theta_n^{bp} \to \Theta_n \to \pi_n/J_n \xrightarrow{\Phi_k} \Bbb Z/2 \to \Theta_{n-1}^{bp} \to 0.$$

Let's start by introducing the cast of characters.
$\Theta_n$ is the group of *homotopy $n$-spheres*. It's smooth manifolds, up to diffeomorphism, which are homotopy equivalent (hence by Smale's h-cobordism theorem, and in low dimensions Perelman's and Freedman's work, homeomorphic) to the $n$-sphere $S^n$. (Actually, we identify $h$-cobordant manifolds. Because $h$-cobordism is now known in all dimensions at least 5, it changes nothing for high-dimensional manifolds; but it explains why $\Theta_4=1$ is possible even though it's an open problem, suspected to be false, that the 4-sphere admits a unique smooth structure. In any case, this is not an important aside.) The group operation is connected sum. The data we're really after is $|\Theta_n|$ - the *number* of smooth structures.

$\Theta_n^{bp}$ is the subgroup of those $n$-spheres which bound parallelizable manifolds. This subgroup is essential, because it's usually the fellow forcing us to have exotic spheres in the other dimensions.

This group is always cyclic (Kervaire and Milnor provide an explicit generator). As a rough justification for this group: the way this goes is by taking an arbitrary element, writing down a parallelizable manifold it bounds, and using the parallelizability condition to do some simplifying algebra until this bounding manifold is particularly simple - at which point you identify it as a connected sum of standard ones, hence that $\Theta_n^{bp}$ is cyclic generated by the standard one. I (or rather, Milnor and Kervaire) can tell you its order: If $n$ is even, $|\Theta_n^{bp}| = 0$; if $n=4k-1$, $$|\Theta_n^{bp}|=2^{2k-2}(2^{2k-1}-1) \cdot \text{the numerator of }\frac{4B_k}{k}$$ is sort of nasty, but in particular always nonzero when $k>1$; and for $n=4k-3$, it is either 0 or $\Bbb Z/2$, the first precisely if $\Phi_k \neq 0$ in the above exact sequence.

$\pi_n/J$, and the map $\Theta_n \to \pi_n/J$, is a bit harder to state; $\pi_n$ is the stable-homotopy group of spheres, $J$ is the image of a certain map, and the map from $\Theta_n$ sends a homotopy 7-sphere, which is stably parallelizable, to its "framed cobordism class". The real point, though, is that this term $\pi_n/J$ is entirely the realm of stable homotopy theory. This is precisely why people now say that the exotic spheres problem is "a homotopy theory problem". (To give the slightest bit more detail: The Thom-Pontryagin construction gives that $\pi_n = \Omega_n^{fr}$, the framed cobordism group, whose elements are equivalence classes of manifolds with trivializations of the "stable tangent bunde". Every homotopy sphere is stably trivial, and the image of $J$ is precisely the difference between any two stable trivializations.) This map $\Theta_n \to \pi_n/J$ might motivate the introduction of $\Theta_n^{bp}$ - since that is, more or less obviously, the kernel. The fact that this map is *not* always surjective - the obstruction supplied by $\Phi_k$ - is the statement that not every framed manifold is framed cobordant to a sphere. I find it somewhat surprising that so many actually are!

The last thing you should know is about the map $\Phi_k$. It's known as the Kervaire invariant. It's known to be nonzero in dimensions $k=1,2,4,8,16$, and might be nonzero in dimension $32$, but that's open. The remarkable result of Mike Hill, Mike Hopkins, and Doug Ravenel is that $\Phi_k = 0$ for $k > 32$. I don't have much to say about this, other than that it's there. Summing up what we have so far:

For dimensions $n=4k-1>3$, there are always exotic spheres coming from $\Theta_n^{bp}$ - lots of them! For dimensions $n=4k-3$, $\Theta_n^{bp} = \Bbb Z/2$ unless $k=1,2,4,8,16,32$. So the only possible odd-dimensional spheres with a unique smooth structure are $S^1$, $S^3, S^5, S^{13}, S^{29}, S^{61}$, and $S^{125}$.

Now to deal with special cases. It is classical that $S^1$ and $S^3$ have a unique smooth structure ($S^3$ is due to Moise); $S^5$ is dealt with by 1) finding a 6-manifold of nonzero Kervaire invariant, showing that $\Phi_2 \neq 0$ and hence that $\Theta_5^{bp}=0$; and then 2) calculating that $\pi_5$, the fifth stable homotopy group of spheres, is zero. You can do this with Serre's spectral sequence calculations. (It was pointed out to me that this means that three different field's medalists work went into getting $\Theta_5 = 1$ - Milnor, Serre, Smale. It is worth noting that there is a differential topological proof, coming from the explicit classification of smooth, simply-connected 5-manifolds, but it isn't substantially easier or anything.)

For $S^{13}$ and $S^{29}$, these are disqualified by the homotopy theory calculation that $\pi_{13}/J$ and $\pi_{29}/J$ are not zero. I do not know how these calculations are done - probably the Adams spectral sequence and a lot of auxiliary spectral sequences, which seems to be how a lot of these things are done. Maybe someone else can shed some light on that.

For $S^{125}$, the paper itself sketches why: There's a spectrum known as $tmf$, and the authors are able to write down a homomorphism $\pi_n/J \to \pi_n{tmf}$ and find a class in $tmf$ that's hit when $n=125$.

So what we know now is that $\pi_{61}/J \cong \Theta_n$. The content of the paper you're talking about is precisely the calculation that $\pi_{61}/J = 0$. The authors access it through the Adams spectral sequence, as far as I can tell (I am a non-expert). Adams SS is notoriously hard to calculate *anything* with - mostly the entire content of the paper is identifying of a *single differential* in the whole spectral sequence. Once this is done, they're able to finish the calculation, but it's hard work. If you want a sketch of how this is done, I found the introduction to their paper readable - see section 3 of the paper.