Prestige bonuses give you boosts in every area after you prestige. If you start with a tap rate of $10k, after you prestige it will be $10k+an amount dependent on that bonus (don't know what it is).

Tech cards give the user boosts of various levels. Some will boost a specific business, giving it a higher amount of profit, some will boost your tap rate or luck amount (i.e. odds of getting certain in-game boosts).

Soldiers are added after you reach a certain amount of profit/prestige bonus after your last prestige time (26 percent, according to an earlier post). The more soldiers you send to the weekly war, the more tech cards you can get back. Those are big boosts, so the game is encouraging you to participate in the war (i.e. to prestige) because you'll be stronger after it.

From the screen you described, you'd click on the Prestige button again if you want to prestige. A big whirlwind comes in and erases all your buildings, and you start over, but stronger than the last time you started. It's a good thing. The game is meant to start you over pretty frequently. Some people on here are saying they get to the point they can prestige after like five minutes. It's taking me around 90 minutes, but I stretch it out so I can keep the Fat Cat (that creepy pimp-looking thing that waves at you and asks you to watch a video to double your profit for four hours). You have to wait four hours for that, regardless of whether you prestige, but it doesn't carry over, which is lame, and should be fixed, or the user should be allowed to click it again after a prestige.

As far as the specific numbers, I have no clue, others are far smarter/more knowledgeable than I.