What causes the electric field to change over series resistors?
You have a choice of models.
Circuit theory will tell you what, but not how or why. Think of your wide wire as several thin wires in parallel, assume the current and voltage conditions are the same for each elemental wire, and use the resistor formulae to add them up.
The Drain Pipe Theory, aka the Hydraulic Analogy, is very intuitive, but of course it's about a totally different system, so it doesn't offer any 'how' into what's going on with electricity. It's surprising how far you can push the analogy before it breaks - potential, current, power resistance, capacitance, inductance, diodes can all be 'explained' with it, you can even build a boost SMPS with it. You only need fat and thin pipes, where the flow through and the pressure across them are proportional. A thin pipe will develop a larger pressure (aka potential) drop for the same flow.
The Drude model is inaccurate, but it's quite intuitive. It sort of works for conductors and resistors, but don't push it too far. It's a classical theory, so doesn't describe anything that needs quantum mechanics for an understanding, like conduction bands and band-gaps. It doesn't tell you how or why anything, and it's not really that good for calculating how much either. It's no better than circuit theory for adding up parallel resistances.
The fundamental theory you need is Quantum Electrodynamics, though quite how photons ping about in DC circuits is beyond me. Perhaps you'd ask this in the Physics stack. Intuitive it's not, and like all physics, it's going to tell you how to calculate the results, but not how or why.