Other people provided lots of references, so I'll just state what I think about the subject.
If you are familiar with statistical physics part of the renormalization, you should already have a good grasp also on QFT renormalization (even if you don't yet know it!). The moral is the same here: divergences arise because our picture is only effective and, more generally, the theory doesn't account for all the realistic effects of the nature (like measurement).
The UV divergences appear because of infinite interaction energies and they point at the fact that the theory might be incomplete (i.e. it's just an approximation of some better underlying theory), so we are not really allowed to take infinite energy limit without somehow modifying our theory to accommodate for this.
What about IR divergence? Well, this limit again cannot be taken if you think about it a little but the reason is different from the UV case. IR limit lets you count with arbitrarily small energies. But is this really physical? What about our measurement? Are we really able to measure arbitrarily small energies? Well, of course not. But QFT doesn't know anything about our measuring devices, so it's not surprising that you again have to account for this by hand.
Another novel point of renormalization is a running coupling. And again, this arises precisely because we started to notice that coupling constants aren't really constants when one thinks more deeply about what it constitutes to measure something.
I think the whole point of renormalization can be stated pretty concisely: it arises because we realized how ignorant we were. Both of ignoring the fact that QFT is not the ultimate theory of everything and also ignoring the subject of measuring.