Physics Videos by Eugene Khutoryansky (a widely respected channel) recently posted a video on YouTube suggesting this. I know that gravitational fields cause time dilation (with time passing more slowly the closer an object is to the centre of gravity, relative to more distant observers), but I'd never heard it suggested that the time dilation experienced in a gravitational field is what causes the gravitational attraction itself. Can someone clarify this for me?

A YouTube video is not an acceptable reference. You will need to find a textbook or peer-reviewed paper. A quick Google search does not turn up any actual papers by this person.

The two phenomena are connected, but that doesn't mean one causes the other. Most general relativists would say that the curvature of spacetime due to a massive object causes both the time dilation and the attraction.

I saw the video by chance and wanted to ask for an opinion on what was being said. It's pretty clear that I wasn't presenting it as some kind of academic reference.

Sorry, 'most general relativists would say' is not an acceptable reference.

And my point was that asking for an opinion on what is being said in a pop science video is not a good strategy if you actually want to learn the science. If you want a quick opinion on what is said in the video, that's it. It's also why PF has rules about acceptable references--because we have enough to do explaining what's in textbooks and peer-reviewed papers, without also trying to explain what's in pop science videos that are often distorted or misleading.

So what? I wasn't asking for your opinion on something someone said. I was simply answering your question. If you want references, check any GR textbook. Sean Carroll's online lecture notes are free and worth reading:

It depends on how fussy you are. The most obvious effects of a gravitational field can be ascribed to "time dilation", i.e the spatial gradient of ##g_{00}##, the time component of the metric, but there are more subtle effects that cannot be put into this framework. There are a total of 16 metric components, looking at only one doesn't give the complete picture. The danger here is that a well-meaning popularization that can provide useful insights when applied properly can be misleading when applied out of context as a complete solution to understanding the entirety gravity.

This is how I see the video. Everybody knows that when you are mowing the lawn and hit the sidewalk at an angle the difference in the resistance between grass and concrete makes the mower change direction towards the grass. Similarly, when light passes through a retarding medium with angled surfaces ( a prism) the right and left differences in the amount of glass as opposed to air makes the light change direction towards the thicker glass. Exotically, as an object is traveling through time in any gravitational field (for instance near the earth's surface) the modestly greater retardation in such progress that any object, even a point particle, experiences on its high and low flanks (as a result of gravitational time dilation) shifts the direction in space of its progress through time. Thus, if unsupported, it falls. Time dilation is responsible for what we experience as gravity in normal low velocity settings. But when something is traveling through space at a speed equivalent to how we are all progressing through time, (like light) then the curvature of space shows dramatically and the total effect, space plus time, is double what we normally experience. If Newton thought light had mass he would have predicted the same starlight curvature around the sun that Einstein mistakenly originally predicted before he figured out the double effect above. In this sense you can think of Newton gravity as the equivalent of time curvature without spce curvature. But weather kept an earlier observation from occuring and in the interim Einstein got it right . Close call. I am without credentials so will now let the physicists have their way with me.

This viewpoint, of the curved spacetime around a gravitating mass as a "refractive medium" that bends the paths of objects towards the mass, is a reasonable heuristic for some purposes, yes.

Not really, because Newtonian gravity (as opposed to the weak field approximation to GR, in which gravity looks very similar to Newtonian gravity) has no gravitational time dilation.

As others noted, the whole "what causes what" approach is pointless here: both are consequences of space-time geometry. The analogy to refraction due to a gradient in propagation speed is OK for the local gravitational 'attraction' (much better than the rubber sheet with balls rolling on it). But there are other effects, like spatial distortions, noticeable over larger areas.

An alternative analogy: Instead of differential advance speed along time axis, use differential distances along the time axis, like in the below video and the links in its description.

The GR metric for Newton's gravity is obtained from the Schwarzschild metric by assuming non relativistic velocities (v^2/c^2 << 1) and weak gravitational field (GM/Rc^2 << 1). In Cartesian coordinates It reads

so it does contain a time deformation term g_00 = (1-2GM/rc^2) (causing gravitational time dilation) as well as a space deformation term (1+2GM/rc^2). For non relativistic velocities, the time deformation is c^2 times larger than space curvature, because of the c^2 term in (c dt)^2.
Therefore, to leading term Newtonian space is flat. And in fact the equation of Newtonian motion derived via a least action principle do not make use of the spatial term of the metric.

Newton gravity originates from the g_00 term, as its spatial gradient (as Pervect wrote), and may be intuitively understood as a differential motions described by Bill Ryan (in fact gravitational force is the gradient of the potential, which in the above metric corresponds to the g_00 term). Space curvature is relevant for relativistic motions, e.g. by doubling the light deflection effect in the weak field limit (whereas for strong gravitational fields the space curvature term becomes dominant). Space curvature is also important for non relativistic motions if the measure is very accurate, e.g. by opening the orbit of planets and causing a precession like that measured for Mercury.

Therefore the visual exemplification of a rubber sheet curved by a heavy ball (the Sun), with smaller marbles orbiting around due to space curvature, is essentially wrong, because Newtonian space is flat. Eugene Khutoryansky's video is thus correct as far as I understand.
The problem is that no GR book discusses this explicitly, but leave the point rather implicit (see for example the discussion on Newtonian metric in Hartle's "Gravity"). Only exception I know is this book: http://www.relativity.li/en/epstein2/read/ by David Eckstein.
Hope this helps a bit.

On the other hand, I think the spatial part of your approximation of the Schwarzschild metric is also important to consider for cases like the famous deflection of light on the sun, because otherwise you underestimate the effect by a factor of 2!

Yes, in fact the spatial term is absolutely necessary when discussing photons or relativistic motions. But discussion of trajectories at relativistic speed is beyond the newtonian approximation by definition.