The professor is basically correct, though they left out some details.
Here is where the 'larger' part comes in: For the same unit projected area (say, the area of your pupils projected onto the light source), an incandescent bulb must produce a lot more flux per unit projected area in the steradian direction being measured to achieve the same illuminance on that same (projected) area as a fluorescent lamp. And yes, it's because the incandescent lamp has less physical area to generate the light from.
Another way to think of it, with less techno-jargon: Luminance is like water through a garden hose. The flux per angle per projected angle is determined by the size of your hose (in this analogy, the size of your pupil). Let's say we have two sources, and one is roughly the same physical size as the hose, and one is way bigger. Both sources can fill a kiddie pool at the same rate (same flux per area). If you put your hose up to the smaller source, you get basically all the flux that was used to fill the kiddie pool. If you put the same hose up to the bigger source, you get less of the flux used to fill the pool.
The main highlight of the hose analogy is that the hose's size doesn't change with distance, which is the whole point of the unit of luminance. You could walk 100m away, but your hose size (defined by your pupil) doesn't change. That's why luminance is a great measure of 'perceived brightness', because our brains don't perceive the brightness of something to change even if we move farther away. It breaks down when you try to think about steradians, but it helps for imagining how much flux makes it through the hose.
As one uses a photometer to measure luminance, I would think that the measurement of each light source would be from a (fixed) distance such that both bulbs fit fully into the angle of acceptance of the luminance meter.
Often, luminance meters use photography-style lenses, with pupils on the order of our eye, focusing light onto small-ish sensors. More likely, in the professor's mind (as in mine - I work for a company that makes luminance meters), the luminance meter was collecting light from a small projected area that was roughly the size of the bulb, but smaller than the lamp.
Addendum about assumptions
Some assumptions are made here, such as the fluorescent lamp has roughly uniform flux emission across its area. The illuminance reaching the surface of interest (say, our pupil) can come from many different points on the fluorescent lamp, so any one of them does not have to emit as much flux as the incandescent lamp. But at a distance of 1m for both bulb and lamp, the projected area of your pupil is the same area on both. The lamp just has more physical area from which to generate flux, and since we also assume the lamp emits roughly equal in all angles, your pupil is receiving flux from all parts of the lamp when adding up the flux per area for illuminance.
Yes, you can create extreme cases where your hose size is WAY larger than your emitting source, and therefore the 'average luminance' in your hose goes way down, since the projected area goes up but the flux remains the same. But this is really not the point of the luminance unit, and why I think the professor was assuming that the luminance meter's size was on the order of the bulb's size. I think the professor was trying to get you all to think about these subtleties.