In the movies, the Death Eaters leave a smoky trail behind them as they fly. As the Order arrive in the Ministry of Magic in Order of the Phoenix, they leave a silvery trail behind them. In the books, there’s no mention of how they’re doing this. And J.K. Rowling said in the books that only Voldemort and Snape could fly like this; first seen in the Battle of the Seven Potters.

Please can somebody explain how the Death Eaters are able to fly like this?

You are confusing the movies (which do have the black and white trails) and the books (which do not have the trails). There is no mention of silvery trails in any of the books. Or black ones.

The air was suddenly full of the swishing of cloaks. Between
graves, behind the yew tree, in every shadowy space, wizards
were Apparating. All of them were hooded and masked. And
one by one they moved forwards ... slowly, cautiously, as
though they could hardly believe their eyes. Voldemort stood
in silence, waiting for them. Then one of the Death Eaters fell
to his knees, crawled towards Voldemort, and kissed the hem
of his black robes.

_{Goblet of Fire - page 561 - Bloomsbury - chapter thirty-three, The Death Eaters}

So, canonically, the Death Eaters don't fly. They Apparate (see quote above).

Regarding the Wikia, it is frequently incorrect in its information, so I would want to know the source of any information gathered there. That said, I checked my copy of Harry Potter: Film Wizardry From the Creators Behind the Celebrated Movie Series by Brian Sibley to see if the black and white smoke trails were discussed. They were not, nor were they even pictured. It is my guess the trails were added for effect by the movie producers to distinguish the good guys from the bad guys, and, frankly, because they're really cool! I think the smoke trails are a great effect -- the Ministry of Magic in Order of the Phoenix the movie are some of the best scenes in the entire series, if you ask me. (YMMV of course!)

As mentioned by Slytherincess, there are no "silvery trails" in the book. As for the flying as smoke in the movies, that was invented for the movies and is ostensibly completely different from Apparition:

From the Harry Potter wikia:

Death Eaters are shown physically collapsing into thick black plumes of smoke, shooting off into the sky, and being able to go wherever they want to, while doing this. Even though many people think this is apparition, it is not. It is just an ability that was given to the Death Eaters in the movie series. Voldemort, for example, can fly this way, and also really apparating in the standard way. That shows that flying (black smoke) and Apparating are different things in the movies.

I'll keep poking around to see if I can find any canonical explanation, but the general consensus seems to be that the movie-makers had money in their special effects budget and/or just thought it looked cooler.

They don’t - that’s how the movies show Death Eaters’ Apparition.

The company that helped design the Death Eaters’ smoky black trails in the movies described it as Apparition when discussing how they created the effects. They consistently describe it as Apparition, and never described it as flight, so this seems to be the movies’ way of portraying the ability the Death Eaters have rather than them giving the Death Eaters the ability to fly in the movies.

Syflex: Where in the film can your SyFlex work be spotted?

David: We used SyFlex in three sequences: the Graveyard, Dark Mark (ministry of magic wizard apparition), and the Portkey.
There were 4 deatheater shots in the graveyard sequence and in total we created 7 deatheater digital doubles that had to be able to be seen very close up on screen. We had the task of creating the deatheater apparation effect. The Death Eaters were to arrive from the sky amongst an amorphous spear of smoke, stabbing into the ground and suddenly taking form amongst tendrils of this acrid black smoke. The black smoke needed to encircle the Death Eaters bodies, becoming the black of their swirling cloaks as they materialized.
_{^{- SyFlex in Harry Potter and the Goblet of Fire}}

They also designed a different Apparition style for when “good” wizards Apparate that’s similar to the Death Eaters’ Apparition, but distinctive enough to set them apart.

In the Dark Mark sequence, the effect we had to create here was very similar to the deatheater apparition. These wizards were 'good' so there needed to be differences in the way the effect played out. SyFlex was used to generate cloth simulations for the wizards as they appeared. This was then used to help drive the smoke that swirls around the actors as they materialize.
_{^{- SyFlex in Harry Potter and the Goblet of Fire}}

What the Death Eaters are doing when they turn into black smoke is consistently described throughout the interview as Apparition.

Syflex: You worked mostly on the Death Eaters... Can you tell us what were the director's requirements here?

David: The Death Eaters had to be dark and scary with a really sinister quality to their apparition effect. It needed to look like an 'amoprhous spear of smoke' stabbing into the ground, with a certain level of supernatural control to the smoke as it swirled around the materializing digital double. The director had shot plates of the deatheater actors as he imagined they would step forward out of the smoke at the end of their apparition, so we knew we could create a transition from a digital double to the real actor if we had to. The problem with this was the takeover point from one to the other - matching the exact position of a digital double to the actor at any particular point would have been very time consuming. In the end the cloth simulation and final render of the Death Eaters was so convincing that in a number of the shots we kept the digital double in the shot and left the actor out altogether.
_{^{- SyFlex in Harry Potter and the Goblet of Fire}}

At no point do they refer to it as flying, or say the Death Eaters have the ability to fly.

Typically it's not flying, it's called unsupported flight, meaning that they are using a spell to Apparate and fly. Their cloaks are what brings out a trail, e.g when Severus attacked McGonagall, as he swooped his wand in the air and shot a spell (invisible spell) and Apparated through the students in Hogwarts.