Even the most apparently reputable sources of information are sometimes full of misconceptions. The BBC manages to demonstrate several all at the same time with this terrible animation trying to explain why thicker wires have a lower resistance than thin ones.
The main argument is that a thick wire has ’more space’ for the electrons to move around in than a thin wire. But wires are made from atoms - that's where the free electrons come from. So thicker wires have more atoms and so no more empty space (per cross-sectional area) than thin ones. Another implication of the animation is that the wires are like empty tubes. This suggests that the electrons come from the battery as a sort of source rather than already being there everywhere in the circuit. The final problem is the speed of the electrons. Since the animation shows a longer path for the electrons in the empty thick wire their speed must have increased. In fact, the opposite is the case. Electrons travel slower in thick wires.
For a copper wire (at a given temperature) the speed of the electrons depends only on the voltage across it. Imagine a three-lane road and a single-lane road with cars all going at the same speed. More cars pass per second in the wider road even though the speed is the same. More cars (charges) per second means higher current for a given voltage and so smaller resistance.
Here's an animation showing how thicker wires have a lower resistance.
Since current is the same around a simple series circuit the charges have to go faster where the wire is thinner. Faster charges mean more interactions with the ionic lattice per second and so higher resistance.
