How to understand adjoint functors?

Nice question Bumblebee. So, let us start with some "metaphysics of adjointness":

THE LEFT AND RIGHT ADJOINTS TO A FUNCTOR

$ \mathcal{F}:\mathcal{C}\hookrightarrow\mathcal{D}$

ARE THE FREE (LEFT) AND CO-FREE (RIGHT) WAYS TO GO BACK FROM $D$ TO $C$.

If you choose some easy examples, for instance $C=Top$ and $D=Set$ and the functor is simply the forgetful functor which "forgets" the topological structure, Left and Right start from a given set and endow it with a topology, in the most economic way (trivial topology) or in the most rigid one (discrete) . Same happens if you replace $Top$ with $Groups$ (or any other algebraic category).

Now, not all functors which have adjoints are forgetful functors, so matters are slightly more subtle sometimes, but the "general metaphysics" of adjointness still holds true.

Now the second part of your question, the scary formula for your example: rather that filling this page with calculations, I want to give you the heuristics (so far I have told you what adjoints are, not whether they exist or how they are calculated).

Here, I use the second "metaphysical principle of adjointness", namely this:

THINK OF CATEGORIES AS GENERALIZED ORDERS, AND OF ADJOINTNESS AS GENERALIZED GALOIS CONNECTIONS.

If you look up Galois connections (see here), how they are defined and calculated, you will also understand cats, by generalizing. Same exact story.....

If you start with a category and only consider what you can see by looking at functors from groupoids, well you’ll only see the invertible morphisms. So the right adjoint is the core.

If you start with a category and only consider what you see by looking at functors to groupoids, well that’s a little harder because a non-invertible morphism can be sent to an invertible one. So to get the left adjoint you have to try to add formal inverses.