Solving something numerically means you use approximations that make a problem easier to solve.
Generally it refers to the difficulty of solving problems mathematically that give you the exact answer, or trying to get approximate answers using techniques that involve numeric approximations, that allow you to get close to the solution sooner or more easily.
For instance, since computers can only store numbers with bits and bytes, many solutions you obtain are only approximations. You cannot store the number known as Pi, for instance. But you can use an approximation of Pi with a large number of bits.
Another example is that certain classes or problems, like partial differential equations, are very difficult to solve. However, you can use methods that are known to give you something that is close to the answer, a numerical approximation. Typically such methods use approximations and computer power. So you never get the exact answer but you get something close to it.
There is a entire field of science known as numerical analysis based on this topic, so my answer cannot possibly be complete, but I’m giving you a very basic take on it.
The tags you used are all related. Numerical integration is what I just described for the problem of integration. Discrete mathematics can be used to approximate infinite or continuous values to get solutions numerically. Numerical methods are a collection of such methods to solve problems numerically.