Louis de Broglie proposed that if light can behave like a particle, then matter can also behave like a wave. He suggested that every moving particle has a wavelength given by $\lambda = \frac{h}{p}$, where $p$ is momentum.

Experiments showed that electrons can create interference and diffraction patterns, just like waves. This was a major breakthrough because it meant that wave behavior is not unique to light.

A matter wave does not mean that a tiny ball is literally smeared like a water wave. Instead, the wave describes the probability structure for where the particle may be found when measured.