Quite simply - you need quantum mechanics because - past a certain point - classical mechanics just doesn’t work.
Classical electromagnetism and thermodynamics was able to prove that when things get hot - they glow.
You’ve all seen this:
As physicists - the next job was to work out how much it is glowing - and at what frequencies? How does this change with temperature?
So, Rayleigh and Jeans came up with a law which said that the spectral radiance (the amount of light emitted at each wavelength) of an object is given by:
${B}_{\lambda }\left(T\right)=\frac{2c{k}_{b}T}{{\lambda }^{4}}$
Where $\lambda$is the wavelength of the light.
Now - as far as classical physics is concerned, this is absolutely the correct answer to have derived.
But.
What happens when $\lambda$ gets small?
Bugger.
Yeah - the theory predicts that the intensity goes off to infinity, as the wavelength goes to zero (the purple/blue line on the diagram).
That would imply that every single object in the universe was constantly spewing deadly X-rays everywhere - all of the time.
This isn’t what happens.
The observed spectrum is the green line - it goes up from zero, peaks, and goes back down. At no point does it shoot off to infinity and bathe the universe with deadly radiation!

This problem stumped a lot of people for a long time - it even had a badass name: “the Ultraviolet Catastrophe”.
To get around this problem, Max Planck postulated that light came in discrete “bundles” - or “quanta”, which had energy $E$ proportional to their frequency:
$E=h\nu =\frac{hc}{\lambda }$
For a bunch of complicated maths reasons (statistical physics is the easiest way to prove this), this added postulate leads to Planck’s law:
${B}_{\lambda }\left(T\right)=\frac{2h{c}^{2}}{{\lambda }^{5}}\frac{1}{{e}^{\frac{hc}{\lambda {k}_{b}T}}-1}$
Which does accurately predict the correct intensities observed:
So that’s why you need it - because our description of nature doesn’t work without it.
Trust me - if physics knew of a decent way to do away with the restrictions that quantum mechanics places on us, then we sure as hell would.
But nobody has found a better theory in 100 years - they just find better and better refinements of our old one.
Quantum mechanics is needed because the universe appears to behave in a quantum fashion - simple as that.