Vaia, why don't we see the problem from a different perspective.
Suppose, there is a box with two strings attached to both sides of it. You and one of your friends are pulling it from both sides. Let, you are pulling it by $F_1$ Newton force, and your friend is pulling it by $F_2$ Newton force. Furthermore, suppose your friend is pulling it by a constant force, i.e, $F_2$ is constant. But, you have the ability to change your force.
Now, in the primary position, say, you're force of pulling is less than that of your friend. Then, $F_2>F_1$. So, the resultant force on the box $F_2-F_1$ will be greater than zero, and, the box along with you will move towards you're friend.
Suppose, you are now gradually increasing your force, whereas, you're friend is still puling at the same force. Then still, the box and you will keep moving towards you're friend as long as your pulling force $F_1$ is
less than you're friend's pulling force $F_2$.
Now, if you come to a position where your pulling force exactly equals you're friends, i.e, $F_1=F_2$, then, the resultant force on the box will be $0$. So, you, you're friend and the box will remain completely still. In this position, if you apply even a
little bit of force
more than you're friend, the box will move towards you.
Now, to come to the actual problem, if you replace you're friend with the Earth, you can easily understand the situation.
But,
how much more do we need to apply???
We will calculate it step by step. Suppose, you applied $1$ Newton force more than you're friend. Then, $F_1=F_2+1$. But, the box would still move towards you if you applied $0.1$ Newton force more, or, even if you applied $0.001$ Newton force more. So, the amount of force you want to apply more
tends to $0$. Take this data to the limit and it will become $0$.
Then, the object will move towards you if $F_1=F_2$. Hence, if the Earth attracts a body with $mg$ Newton force, in order to lift it up, you also need to apply $mg$ Newton force
!!! (
Actually, you need to apply more than $mg$ Newton, but, since
we cannot calculate how much more, we suppose it is $mg$. Ah! Finally finished this post.
)