Why the probability of density is higher in the area that is closer to the nucleus? I'm a high school student. I don't know much about wave functions.
First of all you have to consider that there's an electric force (Coulomb) between nucleus and electron, so, from a classical point of view it seems reasonable that electron "wants" to stay as closer as possible to the nucleus (they respectively attract). Now it's clearly impossible that electrons could reach nucleus, from an intuitive view points you can see it if you apply the Uncertainty Principle: if the position of electron is precisely define (at the origin if we put the nucleus in that position), electron must have infinite kinetic energy and this is obviously impossible. So electron reach the equilibrium position near the nucleus. Despite that it's clear that from a merely classical discussion electron is attract by the positive charge of nucleus so, as a consequence in every theory (QM too), the probability of density is higher near nucleus than far away from it.
The short answer to your question is that in "nature;" energy, forces, and particles, operate in such a way that the energy or force needed to reach some kind of "stability" (equilibrium), is minimized. In the case of an electron, its "ground state" is the most stable, so the probability of "finding" it there, is very high (if not the highest).
For the 1s orbital the highest probability density exists inside the nucleus. Welcome to the subatomic world!
But there are many other orbitals where the density inside or near the nucleus is low. Look e.g. at http://www.chemistry.mcmaster.ca/esam/Chapter_3/section_2.html for details (note that a standard “hydrogen atom” mantra doesn’t signify much of restriction and in other atoms orbitals are not very different.