Welcome to the weird world of quantum mechanics. The question you are asking is the type of question one unavoidably asks as he tries to learn the counter-intuitive concepts of quantum theory. A book you might like, which is written in a simple style, is The Meaning of Quantum Theory by Jim Baggott.

In answer to your question, here are some points which may be found in the book. They approximate de Broglie's thinking. In special relativity, the relationship between E, p, $m_0$, and c is given by $$ E^2 = p^2 c^2 + m_0^2 c^4 $$

A photon moves at the speed of light. De Broglie thought that a photon had a very small rest mass, though we know today that it has zero rest mass. Nevertheless, if you assume that a photon has a vanishingly small rest mass, the equation reduces to $E=pc$. It was known at the time that the energy of a photon is given by $E=h\nu$. Now if you combine these last two simple equations you get $pc=h\nu$. Since the frequency $\nu$ is given by $\nu=c/ \lambda$, one gets $$\lambda=h/p$$

which is de Broglie's result. De Broglie suggested that this equation might work for matter as well as photons. This result was part of his PhD thesis, and his proposal influenced Schroedinger.

Clearly this result does not fit with a classical physical understanding of matter. Instead, it implies that the probability of finding a particle in a particular place will involve the superposition and interference of waves in the same way as light.

You can think of the wave nature of matter not as implying that matter moves in a wavy motion or that there are waves connected with matter, but rather that matter is not fixed in location and that its location (and momentum) is only describable in probabilities due to its wave nature. This is reflected in the uncertainty principle which applies to noncommutative properties such as position and momentum. The phrase "intrinsic to" might be substituted for “associated with.” Wave properties are intrinsic to matter, part of its nature, though that aspect of its nature is invisible in the macroscopic world. It becomes apparent in the sub-microscopic world of quantum mechanics.

The wave nature of matter is often viewed from our macroscopic, every-day vantage point as a property which is dual to the corporeal particle-like nature that we think of matter as possessing. I don’t especially care for that viewpoint since it puts one foot in the macroscopic world and one in the sub-microscopic world. I prefer to think of mass as having an intrinsic wave property. Because this property is normally only visible in the sub-microscopic world, we only perceive matter as possessing particle-like properties in everyday situations. Ascribing particle-like properties to matter is an excellent approximation in the macroscopic world in which we live and move. We don’t need quantum mechanics to play billiards but we do to probe the properties of atoms.

I hope this helps. I am still learning these concepts myself and precise language is needed to correctly describe the situation. Others more experienced than I may give a better (more accurate) explanation.

Welcome to the weird world of quantum mechanics. The question you are asking is the type of question one unavoidably asks as he tries to learn the counter-intuitive concepts of quantum theory. A book you might like, which is written in a simple style, is The Meaning of Quantum Theory by Jim Baggott.

In answer to your question, here are some points which may be found in the book. They approximate de Broglie's thinking. In special relativity, the relationship between E, p, $m_0$, and c is given by $$ E^2 = p^2 c^2 + m_0^2 c^4 $$

A photon moves at the speed of light. De Broglie thought that a photon had a very small mass, though we know today that it has zero mass. Nevertheless, if you assume that a photon has a vanishingly small mass, the equation reduces to $E=pc$. It was known at the time that the energy of a photon is given by $E=h\nu$. Now if you combine these last two simple equations you get $pc=h\nu$. Since the frequency $\nu$ is given by $\nu=c/ \lambda$, one gets $$\lambda=h/p$$

which is de Broglie's result. De Broglie suggested that this equation might work for matter as well as photons. This result was part of his PhD thesis, and his proposal influenced Schroedinger.

Clearly this result does not fit with a classical physical understanding of matter. Instead, it implies that the probability of finding a particle in a particular place will involve the superposition and interference of waves in the same way as light.

You can think of the wave nature of matter not as implying that matter moves in a wavy motion or that there are waves connected with matter, but rather that matter is not fixed in location and that its location (and momentum) is only describable in probabilities due to its wave nature. This is reflected in the uncertainty principle which applies to noncommutative properties such as position and momentum. The phrase "intrinsic to" might be substituted for “associated with.” Wave properties are intrinsic to matter, part of its nature, though that aspect of its nature is invisible in the macroscopic world. It becomes apparent in the sub-microscopic world of quantum mechanics.

The wave nature of matter is often viewed from our macroscopic, every-day vantage point as a property which is dual to the corporeal particle-like nature that we think of matter as possessing. I don’t especially care for that viewpoint since it puts one foot in the macroscopic world and one in the sub-microscopic world. I prefer to think of mass as having an intrinsic wave property. Because this property is normally only visible in the sub-microscopic world, we only perceive matter as possessing particle-like properties in everyday situations. Ascribing particle-like properties to matter is an excellent approximation in the macroscopic world in which we live and move. We don’t need quantum mechanics to play billiards but we do to probe the properties of atoms.

I hope this helps. I am still learning these concepts myself and precise language is needed to correctly describe the situation. Others more experienced than I may give a better (more accurate) explanation.