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The Culture is known for having huge starships that are capable of sustaining millions of people and traveling faster than light.

Do the novels attempt to explain the power source for these ships ?

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Sources of energy for the Culture are fusion, anti-matter (used for small powerful devices like hand guns), and the Energy Grid, the latter for starships, being the most plentiful and complete technobabble, which Banks was always happy to admit. From his essay A Few Notes on the Culture:

Lastly, something of the totally fake cosmology that underpins the shakily credible stardrives mentioned in the Culture stories. (...) We accept that the three dimensions of space we live in are curved, that space-time describes a hypersphere (...) Within our universe, our hypersphere, there are whole layers of younger, smaller hyperspheres. And we are not the very outer-most skin of that expanding onion, either; there are older, larger universes beyond ours, too. Between each universe there is something called the Energy Grid (I said this was all fake); I have no idea what this is, but it's what the Culture starships run on.

Accordingly Banks never went into much detail beyond this. The Grid is both a source of energy and a means of traction: Culture ships use field engines/grid motors to push against the grid and thus propel themselves. From Consider Phlebas, chapter 9:

Imagine a vast and glittering ocean seen from a great height. It stretches to the clear curved limit of every angle of horizon, the sun burning on a billion tiny wavelets. Now imagine a smooth blanket of cloud above the ocean, a shell of black velvet suspended high above the water and also extending to the horizon, but keep the sparkle of the sea despite the lack of sun. Add to the cloud many sharp and tiny lights, scattered on the base of the inky overcast like glinting eyes: singly, in pairs, or in larger groups, each positioned far, far away from any other set.

That is the view a ship has in hyperspace as it flies like a microscopic insect, free between the energy grid and real space.

The small, sharp lights on the undersurface of the cloud cover are stars; the waves on the sea are the irregularities of the Grid on which a ship traveling in hyperspace finds traction with its engine fields, while that sparkle is its source of energy. (...)

The two “Killer” class Rapid Offensive Units Trade Surplus and Revisionist raced through the hyperspace (...) Their engines were each a focus of energy almost beyond imagining, packing sufficient power within their two hundred meters to equal perhaps one percent of the energy produced by a small sun, flinging the two vessels across the four-dimensional void at an equivalent speed in real space of rather less than ten light-years per hour. At the time, this was considered particularly fast.

The Grid is said to be more "tenuous" in the empty space outside the galaxies, making space travel much slower there.

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It’s stated in the books that they use something called the “energy grid“, which seems to be basically the zero-point energy inherent in the vacuum.

Physicists Richard Feynman and John Wheeler calculated the zero-point radiation of the vacuum to be an order of magnitude greater than nuclear energy, with a single light bulb containing enough energy to boil all the world's oceans.

For example, from Excession:

The big GSV was already moving, rolling and twisting to point directly upwards out of the system. Its engine fields reconnected with the energy grid.

Consider Phlebas is set centuries before the other Culture books, and they seem not to have mastered the technology for powering ships at that time:

That line of light was part of the grid itself, the fabric of pure energy which lay underneath the entire universe, separating this one from the slightly younger, slightly smaller antimatter universe beneath. The Culture, like the Idirans, could now partially control that awesome power, at least sufficiently to use it for the purposes of destruction. A line of that energy, plucked from nowhere and sliced across the face of the three-dimensional universe, was down there: on and inside the Orbital.

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    If you have a copy of Excession handy, I think it explains that the energy grid is the "boundary" of the universe, existing beyond infra- and ultraspace and separating the universe from the ones "adjacent."
    – DavidW
    Jun 4 at 3:11
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    Worth noting however that the Culture series is pure space opera, not hard sf: Banks wasn't very interested in providing scientific explanations for how stuff worked. Jun 4 at 8:16
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The appendices of 'Consider Phlebas' has information on Culture energy production when describing the motivation, or lack thereof, for the Culture-Idiran war:

And energy was, if anything, even more generally available (than matter) through fusion, annihilation, the grid itself, or from stars, taken either indirectly as radiation absorbed in space or directly tapped at the stellar core.

So The methods would be:

  1. Fusion - not the most efficient I think, but easy if you don't have a lot of antimatter stored and can grab some hydrogen and helium.
  2. Annihilation - I think this means matter/antimatter annihilation. I don't think antimatter is very plentiful in nature, so it has to be manufactured using some other form of energy. Stored antimatter is however much more powerful than any of the other methods save perhaps the energy grid.
  3. The grid itself - I assume somehow using finely tuned fields to extract energy from the grid (explained below).
  4. Stellar radiation - Culture 'field technology' is ill-defined, but ships can and do project fields with about any properties you can think of. I think they could probably produce fields that would absorb 100% of the stellar radiation shining on them or focus the radiation into something that could absorb a lot. I doubt this would be used for space travel because it wouldn't be as powerful as other methods and because there is not that much stellar radiation between the stars.
  5. Stellar cores - I imagine this would be using wormholes to displace incredibly hot and dense matter from the cores of stars to the ship and using the energy contained therein. Like #4, not very useful between the stars because there is a limited range. This could be used when near stars to produce large quantities of antimatter for later use however.

I don't recall a description in any of the books of a Culture ship running out of fuel, so my guess is that they are able to get all the energy they need from the grid.

Ships do take time to produce munitions such as CAM (compressed antimatter) warheads. My guess is that they are able to use energy from the grid to create antimatter.

In surface detail there is an armada of ships from a level or two lower in the technological hierarch from the culture where it is mentioned that they will work on fusion but be much more powerful if they have time to create their own antimatter, so a higher civilization provides them with some antimatter, but importantly not from their flagship. So fusion is apparently enough for space travel.

The Energy Grid

The cosmological model in the culture universe is explained in 'Excession', the first book from the series that I read. The universe we see is like the surface of an expanded balloon. This balloon resides inside another balloon, which resides inside another balloon, etc. Inside our balloon is another balloon, and inside that is another balloon, and on until supposedly there is some universe creation engine creating these expanding universes like Russian nesting dolls. The three dimensional space and matter in our universe exists on the surface of one balloon. Between it and the next outer balloon is a region of hyperspace called ultraspace. Between it and the next inner balloon is a region of hyperspace called infraspace.

Seperating those areas from their counterparts of the other balloons is something called the 'energy grid'. It is a chaotic energy field. Ships propel themselves through hyperspace by using the fields created by their engines to 'push' against the grid in hyperspace. At one point a ships is described as diving from infraspace through normal space to ultraspace and back again. I imagine the engines are pushing so hard off the lower grid in infraspace that it pushes them to ultraspace where it has to push off the higher grid to increase or maintain velocity in what would look like a sine wave with real space in the middle, hopping back and forth between infraspace and ultraspace.

The energy grid can be used as a weapon, something that happened in the Idiran war was using those engine fields to distort the grid and cause ripples large enough to penetrate real space where they would annihilate anything they came in contact with.

I assume getting energy from the grid entails those finely tuned engine fields somehow.

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