Let $X$ be a projective smooth connected curve over $\mathbb{C}$, let $D \in \mathrm{Div}(X)$ be a divisor and let $p$ be a point of $X$. Consider the exact sequence of sheaves over $X$ $ 0 \to \mathcal{O}_X(-p) \to \mathcal{O}_X \to \mathcal{O}_p \to 0, $ where $\mathcal{O}_p$ is the skyscraper sheaf that is $\mathbb{C}$ on the point $p$. Tensor with the invertible sheaf $\mathcal{O}_X(D)$ and you get another exact sequence of sheaves over $X$: $ 0 \to \mathcal{O}_X(D - p) \to \mathcal{O}_X(D) \to \mathcal{O}_p \to 0. $ Taking global sections, you get an exact sequence of $\mathbb{C}$-vector spaces $ 0 \to \mathcal{L}(D-p) \to \mathcal{L}(D) \to \mathbb{C} $ that implies $1 \geq l(D) - l(D-p)$, as you want.
The only difficult point is to understand the (not canonical) isomorphism $\mathcal{O}_p \otimes_{\mathcal{O}_X} \mathcal{O}_X(D) \simeq \mathcal{O}_p$ in terms of rational functions.
EDIT. Now I am giving a proof without mentioning sheaves, as required by algeom. This is substantially a translation of what I wrote above. Let $K$ be the field of rational functions of $X$. It is well known that Weil divisors are locally principal (this follows from the fact that $X$ is locally factorial), hence there exist an open neighborhood $U \subseteq X$ of the point $p$ and a rational function $\phi \in K^*$ such that $D \vert_U = \mathrm{div}_U(\phi)$ as divisors over $U$. Now consider the the map $\alpha \colon \mathcal{L}(D) \to \mathbb{C}$ defined by $ \alpha \colon f \mapsto (f \phi)(p), \quad \forall f \in \mathcal{L}(D). $ This is well posed because the rational function $f \phi$ is regular in $p$. You should be able to prove that $\mathcal{L}(D-p) = \ker \alpha$. Conclude applying rank-nullity theorem to $\alpha$.