A takeaway in football is when possession of the ball is switched between teams as a result of a turnover. In simpler terms, the defense "takes" the ball away from the offense resulting in a change of possession.
The blanket definition for a fumble is to lose possession of the football, leaving it vulnerable to be recovered by the opposing team. While there is the possibility the other team grabs the ball (takeaway), that does not need to occur for it to be ruled a fumble. Fumbles can happen in a couple different ways.
The defender punches at the ball jarring it loose from the offensive player's arms.
Alternatively, the offensive player could simply drop the football or make a bad pitch to a teammate leading to the ball being on the ground.
Takeaways can swing a game in favor of one team very quickly. They can help a team come back and overcome a deficit, blow a game wide open, or seal the victory.
A takeaway near the opposing team's endzone sets your team's offense up with a short field. A takeaway near your endzone kills the momentum of the other team and essentially takes points away from them. Both of which benefit a team greatly.
Teams talk about winning the "turnover battle" in each game they play in. It is a part of every team's game plan, to have more takeaways than the other team. Doing so gives you a much higher chance of winning the game.