Imagine you start at the front door of a house, you go on a hike through curvy terrain, and then you come back to the same door where you started. As you went on that walk-about and came back to the same point, all the ups and downs summed up to zero. You are at the same altitude as when you started. Here you have a simple reasoning for Kirchoff's Voltage Law, and why the voltages sum up to zero.
Now imagine you and a friend start at work, but before lunch, you need to go the bank, and your friend needs to go the post office. You each take different routes, but for lunch you meet at the cafe, and after lunch you go back together. You have both completed your circuit, and are back where you started. Now imagine that these two trips took place in two parallel worlds where you are each all alone. Would the total amount of people coming back from the cafe be any different ?
How the mesh current method works is basically like that. Where the mesh currents meet you sum the contributions.
In linear circuit analysis mesh analysis (and nodal analysis) almost always works as a method to provide a single solution. In part I think it is helpful to look at the superposition principle. The end-game is anyway that for a linear circuit mesh analysis provides N equations for N mesh currents, with no unknown variables (all mesh currents were included). From linear algebra we know that N equations is exactly what we need to solve N variables.
However a couple of times mesh analysis can fail. Like in linear algebra, if our set of equation has a zero determinant, then there may be many (infinite) solutions. A simple way of achieving a circuit like this is to hook up two equal voltage sources in parallel. How much current goes from one source into the other ? In theory it could be anything.
Another case that gets simple mesh analysis into trouble is when there is no way to "flatten" the circuit into a 2D planar circuit. E.g. consider adding a resistor between A and B in the drawing below. How would you describe the mesh current in the other elements between A and B? For this there is loop analysis.
However a linear circuit is only an approximation and it can meet it's limits. Say that voltage over a capacitor far exceeds it's voltage rating - it would break down - it might even explode - and thus completely change your circuit. That's a bit like you and your friend crossing a bridge on your way back to work, and 10 million other people happened to be on the same bridge (it breaks down!).
Non-linear elements also quickly make trouble for mesh analysis as a general method. By this I mean that mesh analysis may not be able to provide you with a solution anymore, although the mesh equations it provides are not wrong per se. With non-linear elements the set of mesh equations can fairly quickly become analytically unsolvable, there could be multiple solutions, unstable solutions, or no solutions at all...