Signs come from the direction you walk

Choose a walking direction around the loop and draw it on the diagram. Then take the elements in order:

1. Resistor crossed along the current arrow: a drop, $-IR$.
2. Resistor crossed against the current arrow: a rise, $+IR$.
3. Battery walked from its negative terminal to its positive one: a rise, $+\varepsilon$.
4. Battery walked from positive to negative: a drop, $-\varepsilon$.

"Batteries add and resistors subtract" survives exactly as long as every battery faces the same way and every guessed current is correct. Put a second source in facing the other way and it fails immediately: walked from plus to minus, that source contributes $-\varepsilon$.

Either choice of walking direction gives the same physics. So does a guessed current direction that turns out to be backward: solve the equations and the current comes out negative, which is the algebra telling you the arrow was drawn the wrong way. There is no need to go back and fix it.

The rule holds for any closed path in the circuit, including one that never touches a battery. Walk down one parallel branch and back up the other:

$$\Delta V_1 - \Delta V_2 = 0 \qquad \Longrightarrow \qquad \Delta V_1 = \Delta V_2.$$

That is where the familiar fact comes from: parallel branches share one potential difference. It is not a separate rule, it is the loop rule applied to a loop with no source in it.

Two consequences worth stating. A circuit with several loops usually needs more than one loop equation, and skipping the source-free ones leaves you with fewer equations than unknowns. And there is no requirement that the drops around an inner loop add up to the emf: they add up to zero, which is what a closed walk always gives.

Sketching electric potential against position around a loop is the loop rule drawn rather than written, and it catches errors quickly. Three requirements:

1. Ideal wires are flat. No resistance means no potential change, so a connecting wire is a horizontal segment.
2. Each resistor step is $IR$ for that element. Bigger resistance, taller step, at the shared series current.
3. The graph ends where it began. Rises and drops cancel over the full trip, because a charge that returns to its starting point has returned to its starting potential.

A sketch that finishes below where it started has one step with the wrong sign or the wrong size, or has drops drawn along wires that should be flat. Checking the closure is the fastest audit of a multi-element loop there is.

Potential is energy per unit charge, and it is a property of position. Take a charge all the way around a loop and it comes back to the same place, so it comes back to the same potential. That is the whole derivation, and it explains why the rule needs no source to be present.

The energy accounting behind it: the source does work on each charge, moving it from low potential to high, and the resistors take that energy back out as internal energy. Around a full loop the two have to balance exactly.

A practical note that follows. The loop rule constrains the potential differences, never the absolute values, which is why grounding a point in a circuit is free: it fixes an arbitrary reference and changes no measurement anyone makes.