使出吃奶的劲儿It's a very informal expression. It's like to say an infant does all it possibly can to get its mother's milk, sucking at the breast as hard as it can. It's really not easy for a little baby to do such things, so this expression is kind of desperate. So it means......strain every muscle, use every ounce of one's strength. With给我(for me), it's like an order, an imposition. Soldier: Sir, it's like mission impossible, please...... Officer: No way!!! You have to make it!!! 给我使出吃奶的劲儿来!!!

spare no efforts to do something, when you try to do weightlifting, you should 使出吃奶的劲儿.

婴儿吃奶的时候，脸紧绷着，好像很使劲儿的样子。 所以用“吃奶的劲儿”来比喻，尽最大的努力来做某事。

“try your best !!”"do as most and best as you can!"

尽最大的努力去做某一件事