In Serge Lang's Introduction to Differentiable Manifolds he says that a function $f:U\to F$ is differentiable at a point $x_0\in U$ if there exists a linear map $\lambda$ of $E$ into $F$ such that, if we let $$f(x_0+y)=f(x_0)+\lambda y+\varphi(y)$$ for small $y$, then $\varphi$ is tangent to $0$.
Tangent to $0$ is defined as follows:
A real valued function of a real variable, defined on some neighborhood of $0$ is said to be $o(t)$ if $$\lim_{t\to 0}o(t)/t=0.$$
Let $E,F$ be two vector spaces, and $\varphi$ a mapping of a neighborhood of $0$ in $E$ into $F$. We say that $\varphi$ is tangent to $0$ if, given a neighborhood $W$ of $0$ in $F$, there exists a neighborhood $V$ of $0$ in $E$ such that $$\varphi(tV)\subset o(t)W$$
What is the intuition behind defining it in this manner?
Lang is trying to say things in a way that also works in infinite dimensional spaces. There, not all norms on a vector space are equivalent, so he is avoiding a definition using metrics on $V$ and $W$, and assuming only that each vector space has a topology. For finite dimensional spaces it is the same as the definition that does use metrics (and independent of the metric chosen, since all such induce the same topology).