In the 1970s, a new dance craze began in The Bronx: B-Boying (known in the media as "breakdancing"). This was a highly athletic form of dancing, and some of the moves needed footwear which would grip the floor. Here is an example of BBoying from the early 80s, by which time BBoying had already been around for a number of years. The need for the dancers to wear gripping footwear should be obvious.

The scene was limited to New York City at first, but grew in popularity. BBoying was associated with "electro" music, which later came to be known as "hip hop", and BBoys were a fixture at hip hop gigs. In those days, a hip hop group was not seen as a real group unless they had BBoys or BGirls who supported their musical performances with dancing.

Many rappers adopted the syles of the BBoys, wearing sports gear and especially the footwear. See, for example, the early LLCOOLJ track "I'm Bad", where he and his group are wearing tracksuits and sneakers. This was a typical hip hop style at the time.

Even when tracksuits were not on display, the footwear was often seen. For example, see this Kool Moe Dee video, "How Ya Like Me Now?" which from the start features dancers in tuxedos and sneakers. By the way: that song is a diss towards LLCOOLJ, and they had a battle over it - but that's another story.

Anyway, the presence of the sneaker was now a part of hip hop fashion. Different groups liked different sneakers. KRS One, for example, was famous for sporting only Nike trainers, and his lyrics reflect this:

How many people got Nikes on?

If you got your Nikes on, put your feet up in the air

If you don't got Nikes on

I think you need to keep your feet down

From "Nervous", 1987.

But what really made it hit the public consiousness was when one group made a song specifically about their sneakers. Run DMC were one of the first hip hop bands to get any attention outside of the hip hop community, and when they released performed "My Addidas", at Maddison Square Gardens to a crowd of thousands of people (mostly wearing Adidas, and some waving them in the air) and Adidas offered them a sponsorship deal the relationship between hip hop and footwear was cemented.

Run DMC had already been popularising the "street" look for rappers on their first two albums, moving away from the previous "glam" look of previous rappers and DJs such as Grandmaster Flash and the Furious Five, and Africa Bambaataa and the Soul Sonic Force towards much more what the fans were wearing: jeans and tracksuits with sneakers.

Run from Run DMC later spoke about the fact that - at the time - they gave no thought to advertising for Adidas. They were just talking about their favorite brand of sneaker, in a song - as was normal for them to talk about any everday subject. The Adidas marketing department, however, did not pass up the opportunity to give them sponsorship, and their own line of sneakers.

Since much of the hip hop community at that time was African-American and Latino, the fashion for hip hop was a big influence on the fashion in those communities.

So the short answer is that sneakers were fashionable among BBoys and BGirls for practical reasons, and that fashion spread throughout the hip hop community in the 1980s. And yes: targetted advertising was beiong done - at first by individual rappers and groups, but later by sportswear manufactuers themselves.

Sources:

"The Freshest Kids: a history of the BBoy" (documentary)

"The Hip Hop Years" - Alex Ogg & David Upshal

EDIT oops got the timeline of Adidas wrong