As per the Bohr’s theory, the electrons cannot occupy arbitrary positions in the atomic space surrounding the nucleus but only exist in predefined discrete levels. Now consider what happens when two or more atoms come closer and each has their discrete energy levels. Since the electromagnetic forces will come into play as the distance between the atoms decreases, the energy levels will get affected by this mutual interaction.
As the inter atomic distance goes on decreasing, the influence of these forces increases and the net result is that instead of discrete energy levels, we now have a narrow range of energy band corresponding to each energy level, and these bands are known as energy bands.
If you find this concept a bit confusing, just refer to the diagram shown here which clearly shows the energy levels labeled as K, L and M on the left hand side of the diagram, whilst the right hand side shows the energy bands corresponding to these energy levels; the inter atomic distance being shows on the X-axis on the left hand side. You will also notice that the M level band is broader than the inner band and so on. This is obviously due to the fact that as the atoms start to come closer, the valence level gets affected first due to mutual repulsion between electrons of both atoms, while the inner levels are affected to a relatively lesser degree. Hence the variation in the outermost levels is more than the inner levels, hence the explanation for this observation. In case you are not fully familiar with the energy levels K, L, M etc just refer to the article on energy shells.