応援する is an expression that can have a lot of different meanings. The most basic meaning is simply stating that you hope something goes well. This also means that you can use 応援する even when talking to people you have just met.
将来{しょうらい}の夢{ゆめ}は何ですか? Do you have any dreams for the future?
カフェやりたいです。I want to run my own coffee shop.
いいですね!私も応援{おうえん}しているよ! That's nice! I'm rooting for you!
As you can see this is just random small talk, and the person obviously has no intent to help with the coffee shop.
このカフェにいつも行ってお金{かね}を投入{とうにゅう}している本当{ほんとう}の理由{りゆう}は
このカフェを応援{おうえん}しているからなんです。
The real reason why I always spend my money in that coffee shop is because I am supporting them.
This example shows that 応援する is not limited to humans, you can support an ideal, a movement or a restaurant. There is also actualy financial support involved. It is natural that you have to pay for your own bills in a coffee shop, but it is implied that the speaker goes to that shop more often that he normally would, just so that he can give them more money.
The last common use is being a fan of someone, for example a musician or idol. This can involve buying CDs, going to concerts, helping them out with their work, or even just regularly encouraging them on social media.
This is why most official announcements end in something like this.
応援{おうえん}宜{よろ}しくお願{ねがい}いします。I am counting on your support!
引{ひ}き続{つづ}き、応援{おうえん}よろしくです! Please keep on supporting me!
There is no way to truly explain 応援, but this should give you a general idea about how to use it.