Excellent question! As far as I know, there is nothing like a rigorous justification that quantum gravity effects should kick in at the Planck mass/length/time. But there is an intuitive justification, which goes something like this:
As you may know, in special relativity, many quantities can be expressed as a power series in $\frac{v^2}{c^2}$.
$$X = X_0 + X_1\frac{v^2}{c^2} + \cdots$$
When you are working with a physical situation where $\frac{v}{c} \ll 1$, you can neglect the higher-order terms in this expression. This means that $c$ is, in a sense, a characteristic speed at which relativistic effects (the higher-order terms) make a significant difference. Of course this doesn't mean you can completely ignore relativity at speeds less than $c$, but a good rule of thumb is that relativistic corrections are insignificant at speeds less than about $c/10$, which is only one order of magnitude off. In our everyday experience, eight orders of magnitude less than the speed of light, that one factor of 10 doesn't really matter that much.
A similar thing occurs in quantum mechanics, although not quite as clear-cut. Basically, quantum effects include things like interference and diffraction of wavefunctions, which become significant when the wavelength is comparable to the sizes of the objects involved. (This is a result of classical wave mechanics; I won't get into the details here.) So you can think of a quantity like $\frac{\lambda}{r} = \frac{h}{rp}$ playing a role similar to the one $\frac{v^2}{c^2}$ does in SR: it gives you an order-of-magnitude estimate of the scale at which quantum effects come into play. It's a very blunt estimate, though, because quantum corrections take many different forms - think of quantum mechanics as giving a power series in "$\frac{h}{\text{stuff}}$," not necessarily $\frac{h}{rp}$ exactly. In practice, it often turns out that $\frac{h}{rmc}$ is a better parameter to use; $\frac{h}{mc}$ is called the Compton wavelength.
Finally, the same thing can be said of general relativity, specifically the part of it that deals with strong gravitational fields. In this case the relevant parameter is $\frac{2Gm}{rc^2}$, which pops up in various exact solutions to the Einstein field equations. This means that the Schwarzschild radius $\frac{2Gm}{c^2}$ is a characteristic radius at which general relativistic effects become significant.
Now, a quantum theory of gravity should combine all of these things. So a generic quantity might be calculable in quantum gravity as a multivariable power series like this:
$$\begin{align}X &= X_{0,0,0} + X_{1,0,0}\frac{v^2}{c^2} + X_{0,1,0}\frac{h}{rmc} + X_{0,0,1}\frac{2Gm}{rc^2} + \cdots \\
&= \sum_{m,n,p} X_{m,n,p}\biggl(\frac{v^2}{c^2}\biggr)^m\biggl(\frac{h}{rmc}\biggr)^n\biggl(\frac{2Gm}{rc^2}\biggr)^p\end{align}$$
where $X_{0,0,0}$ is the classical, low-speed approximate value, and the subscripts refer to how many powers of the various correction factors are involved. Any term which has zeros for certain combinations of the subscripts, we can already calculate using existing theories. For example, calculating some quantity using special relativity gives us the power series
$$X_\text{SR} = \sum_{n} X_{n,0,0}\biggl(\frac{v^2}{c^2}\biggr)^n$$
which takes care of all the $(n,0,0)$ terms. Similarly, quantum mechanics takes care of all the $(0,n,0)$ terms, and GR all the $(0,0,n)$ terms - in fact, because GR includes special relativity, you can also get the $(m,0,n)$ terms from it. The terms with indices of the form $(m,n,0)$ come from the combination of special relativity and quantum mechanics, namely quantum field theory.
So what's left? The remaining terms, which aren't covered by existing theories, are going to be negligible unless we have a system that is:
- very dense, $r \sim \frac{2Gm}{c^2}$
- and very small, $r \sim \frac{h}{mc}$
Equating these conditions, we get $\frac{2Gm}{c^2} \sim \frac{h}{mc}$, or
$$m \sim \sqrt{\frac{hc}{2G}} \sim m_\text{Pl}$$
(I dropped a factor of $\sqrt{\pi}$ because we're just working in orders of magnitude), and plugging back in to either of the conditions,
$$r \sim \frac{h}{m_\text{Pl}c} \sim \ell_\text{Pl}$$
So the kinds of systems where the $(0,m,n)$ terms in this "master power series" become relevant are precisely those where the mass is on the order of the Planck mass and the size is the order of the Planck length. For calculating the behavior of these systems, we will need a theory which allows us to calculate those $(0,m,n)$ terms, and in fact all the terms with arbitrary indices - in other words, a quantum theory of gravity.
