The atomic radius is a measure of the size of its atoms, the distance from the nucleus to the boundary of the surrounding cloud of electrons.
It decreases as you go left to right because of the greater amount of protons. The higher the atomic number, the greater number of protons. The protons will pull the electrons closer to its nucleus thus resulting in a smaller atomic radius.
It increases as you go down the group because as you go down the group, you add another principal energy level. The more energy levels, the further the electrons will be from the nucleus resulting in a a smaller atomic radius.
Picture a circle and what you know about the radius of a circle. It helps. ;)
The atomic radius is actually really tough to define well. But simply, an atom that is not bonded has an electron cloud that is pretty much spherical. The atomic radius is the radius of that sphere, think of it as the distance from the center of the nucleus to the outer electronic orbital. [ However, the electron shell is really a three dimensional thing with decreasing density toward the outside. The radius can be measured by throwing things at the atom (protons, electrons, photons) and measuring at what distance from the nucleus they get deflected by the atom. Depending on what you throw and what energy (how fast) you throw it, you get slightly different answers. ]
Be that as it may, the radius gets smaller across the periodic table row because the row represents all the atoms with electrons in a given shell, with the next row progressing to the next larger shell. So in one row the basic shell 'size' is about the same, but as you go across the charge in the nucleus increases. The extra attractive charge in the nucleus pulls the electrons in the shell closer, decreasing the size. As you go down the group, you graduate to higher energy shells that are farther from the nucleus. That is simply the intuitive concept that more electrons fill more space.