Answer
The evidences that led to the introduction of quantum mechanics are
$i)$ Black Body Radiation.
$ii)$ Photo Electric Effect.
$iii)$ Atomic Spectra Hydrogen.
$iv)$ Stability of atom.
Work Step by Step
Until the beginning of $20^{th}$century, we understood the properties of materials, molecules or any form of matter that we use in our daily life, using classical mechanics. But in the beginning of the $20^{th}$century when people could look at materials at microscopic level or when they started dealing with atoms, molecules and sub-atomic particles, they realize some strange properties. These properties or behaviors of matter was completely new and could not be explained by any theory so fast served us.
Let's see the properties/evidences one by one that led to the introduction of quantum mechanics.
$i)$ Black Body Radiation: The classical theory of matter-light interaction predicts that the energy emitted by a black-body will diverge with the increasing frequency. However,the experimental results were looking very different. The experiments showed that the "wavelength maximum" of the emitted electromagnetic radiation actually depends on the temperature of the black body.
$ii)$ Photo Electric Effect: When light beams are impacted on metal surfaces, electrons are released. One can say that the kinetic energy of the emitted electrons are related to the energy of the incident light. According to classical theory, the energy of light beam is directly proportional to intensity($I$) which was proportional to square to frequency $v^2$
Hence $K.E \propto I \propto v^2$
But experiment predicted that the energy of the emitted electron was varying linearly with frequency $v$
Also classical theory predicted that electrons would be emitted at any value of $v>0$ but experiments showed that electrons are emitted only after certain threshold frequency $v_0$.
$iii)$ Atomic Spectra Hydrogen: Classical theory predicted that atoms should emit energy continuously but it was observed that emitted wavelengths were discrete.
$iv)$ Stability of atom: Classical theory could not explain the stability of Atom.
Hence, this all lead to the dawn of Quantum Mechanics
