Mass, as you know, is a physical property of all material objects. Historically, mass was defined as a measure of resistance to a change in motion; the more massive an object, the greater the force needed for the object to acquire a given rate of change in velocity (acceleration) This kind of mass is what is known as 'Inertial Mass'. As you also know, all matter is made up of what is known as 'elementary particles', which form a relatively small set. Ordinary matter is made up of protons, neutrons and electrons, each one with a definite value of its amount of mass. Therefore, you can also think of mass as a measure of the amount of matter a given object has. So far, I have managed to give an answer of the form "Objects have mass because they are made of particles, each having mass", so this may not be very helpful. On the other hand, no one knows yet why each elementary particle has mass or why it has the particular amount it has. One possible answer is given by the hypothetical existence of a particle known as the Higgs Boson, which would explain the origin of mass in particles. Of course, this would change your question to "Where does this Higgs Boson come from"?, and this could go on forever (which is typical whenever you find yourself in pursuit of knowledge).

Another property associated with the mass an object has is how it responds to a gravitational field -such as that provided by planet earth- which gives rise to the concept of weight. The weight of an object in a given gravitational field depends ONLY on the amount of mass it has. It doesn't depend on the material it is made of, the shape it has, not even on the date of manufacture (no expiration date here). When using mass in this fashion, you refer to it as 'Gravitational Mass'. This is very remarkable if you think of it. Galileo was the first person to realize this, and then Newton capitalized on it when he gave us his famous law of Universal Gravitation. Einstein went even farther when equating this idea of gravitational mass with the original idea of inertial mass. This gave rise to his General Theory of Relativity. Einstein also found that you can convert energy into mass and viceversa, through the famous equation E = m c² . So, in order to create mass, you need ("massive" amounts of) energy. This brings your question to the Cosmology realm. In the beginning, the universe was all energy and some of it converted into the particles we know (and some we may not know about yet), each endowed with a particular value of mass (according to which kind of particle it is, e.g. each electron has a mass value of 0.000,000,000,000,000,000,000,000,000,000,911 kilograms, regardless of where it came from). This happened in the first 3 minutes in the history of the universe. Most of the elementary particles that make up the objects you are familiar with, come from that era.

I know I haven't yet answered your question, so here it is. The straight forward answer to your question is "Nobody knows (yet)", but I thought that giving what is known as an "Operative answer" may help you better understand the question itself. We don't know why objects have mass, but I think it is more important to first know what mass is and what it does.

Hope this helps. If not, feel free to write again.

You might as well ask why objects have energy; the two are equivalent, as Einstein calculated. You can't really ask in physics why certain things in the universe have properties, particularly something as ubiquitous as mass or energy; they do, and that's about all that science can say.

A more interesting question from the physics standpoint is why the mass of objects inertia is always the same as the mass of the object's gravitational strength. One suggestion is that inertia is somehow the product of the object interacting gravitationally with the entire rest of the universe. However, this is just wild speculation; we don't know.