Intuition behind Lie algebra being an "infinitesimal transformation"

Question

I have been having some trouble intuitively understanding the "infinitesimal" behavior of a Lie algebra. Currently, I think of Lie groups intuitively being a group of continuous transformations. For example, any element of the Lie group $SO(3)$ corresponds to rotation about some angle $\theta$ in $\mathbb{R}^3$. However, I do not know how to interpret the corresponding Lie algebra $\mathfrak{so}(3)$. Based on what I have read any element in $\mathfrak{so}(3)$ is an "infinitesimal" rotation about some angle $\theta$. But what does this mean exactly?