What is the idea behind the renormalization group as it is used in quantum field theory and statistical physics/condensed matter physics? And what kinds of problems can be addressed/solved by using it?
The main idea behind renormalization is the scale invariance of the essential physics.
Shankar's review is a rather good introduction into the subject. In particular, he makes an interesting point about the interplay between the field theories, which suffer from infrared and ultra-violet singularities, and the solid state physics where such singularities do not exist, due to the presence of natural cutoffs. Renormalization group is then essentially studying of how physical properties change with the size of the cutoff, and finding which of the are invariant to the cutoff.
The basic idea, in very general terms, is explained in
In the notations of that post, and regarding the RG in QFT and statistical mechanics, $E$ is theory space, e.g., the set of all models say living on the unit lattice. For such a model $v_0$, like Ising at the critical temperature and zero magnetic field, a feature of interest $Z(v_0)$ could be "the long distance behavior". If this sounds too vague, then take the exponent of the power law governing the decay of the two-point function. One way one can define an RG map $E\rightarrow E$, is using the block spin procedure, i.e., averaging over blocks of linear size $L$ and multiplying by a suitable power of $L$.
In statistical mechanics, one usually has a given point of interest $v_0$, and the use of the RG amounts to studying the trajectory starting from $v_0$. In QFT, it is more involved because one has a sequence of starting points which depend on the UV cutoff (e.g., $\phi^4$ model with cut-off dependent couplings), then one iterates the RG a number of times dependent on the cutoff up to say some fixed "anthropic" scale. The requirement is then to make sure the obtained effective theories, at this fixed scale, converge when the cutoff is removed. I will not repeat here the details which can be found in this other answer