The light may dim with distance from which its coming But how feeble may be infinite of them should make a cumulative effect of the glowing sky.
The main reason is dust in space that absorbs the light. For instance, we are in a great big spiral galaxy. Generally, if you look at a spiral galaxy, you see an immense, bright core. So why don't we see such a core? It's because of the dust. The core, if visible, should be right around the constellation of Sagitarius. But since there's so much dust in the plane of the galaxy, we can't see it in the invisible spectrum. Astronomers have managed to infer the direction of the center of the galaxy by looking for a type of light (21 cm radiation) that is regularly produced by astronomical phenomena, and which has an ability to pass more easily through the clouds of dust in space. By gauging the direction of most of this radiation, they can infer that the bulk of the galaxy is in the direction of Sagitarius.
And really, there is "light" coming from all directions, but most of it isn't in the visible range. Cosmologists have found what is referred to as the 3K (three "degrees" Kelvin - a very cold temperature) radiation background that exists in direction. This is seen not as the product of stars, but as a remnant of the big bang.
But most importantly, there are not really "infinite" stars. The universe, though boundless, is not truly infinite. There are some mathematical oddities that you find when you try to deal with the infinite and the boundless. For instance, there are an infinite number of positive integers, and there are an infinite number of irrational numbers between 0 and 1. As odd as it seems, there are actually more irrational numbers between 0 and 1 (what my math professor called the "uncountably infinite") than there are positive integers (so-called "countably infinite"). Due to the relativity of simultaneity shown by Einstein, one cannot exactly quantify the total number of stars in the universe at a given point in time, because an observer moving at a different speed and looking at that moment in time would disagree as to which stars had already exploded or not, and which stars had already formed or not. So while you can really make a definitive statement as to the number of stars, the number is not really infinite in the general sense of the word, as neither is the universe truly infinite.