Why does regularization allow least square to be solve in closed form?

For example say that you have (in latex)

$$\min_u \|Gu-g\|_2^2$$ 

If it is full rank then

$$2G^T(Gu-g) = 0 \Rightarrow U=(G^{T}G)^{-1}G^{T}g$$ 

If it is not in full rank

$$\min_u \|Gu-g\|_2^2 + \lambda\|u\|^2_2$$ $$ U=(G^{T}G + \lambda I)^{-1}G^{T}g $$ 

Why does regularization helps at all, any explanation would be appreciated.