How do electric fields in circuits exactly work? I have been introduced to circuits and told that an electric field, along with a emf provided by an external source (which can't be electrostatic), causes electrons to move. They follow the direction of lower potential,and then this emf "pushes" them back to the original potential, like going down a slide and then having someone take you to the top again.
However, I don't really have an intuition as to how this electric field would be generated . How can the source create a field that magically follows an arbitrary direction of wire? Or how does it adapt if we twist the wire or move it, for instance? Should I consider the field created by the rest of electrons as well in order to account for it?
 A: 
How can the source create a field that magically follows an arbitrary direction of wire? Or how does it adapt if we twist the wire or move it, for instance?

The important thing in this regard is the distribution of charge on the surface of the wire. Inside the wire the E field is mostly along the wire, and outside the wire the E field can vary substantially in direction relative to the wire. The surface charge is responsible for the often sudden change in the E field inside vs outside the wire.
My favorite paper on this topic is this one: https://www.tu-braunschweig.de/index.php?eID=dumpFile&t=f&f=138440&token=2cc8a71e4fdbf159121c6b8ef8348952a2e0c197 (R Muller. Am. J. Phys., Vol. 80, No. 9, September 2012)
It describes in a graphical semi-quantitative manner how to determine easily where surface charges will accumulate to provide the necessary E fields for the current to take the correct path.

They follow the direction of lower potential

As a side note, it is important to remember that this is a statement that is only true in Ohmic materials and circuit elements, like resistors and conductors. It is not true in general, and it is frequently violated. Do not hold on to this concept too rigidly.
