Graph representation learning (GRL) has emerged as a powerful technique for solving graph analytics tasks. It can effectively convert discrete graph data into a low-dimensional space where the graph structural information and graph properties are maximumly preserved. While there is rich literature on node and whole-graph representation learning, GRL for link is relatively less studied and less understood. One common practice in previous works is to generate link representations by directly aggregating the representations of their incident nodes, which is not capable of capturing effective link features. Moreover, common GRL methods usually rely on full-graph training, suffering from poor scalability and high resource consumption on large-scale graphs. In this paper, we design Subgraph Contrastive Link Representation Learning (SCLRL) -- a self-supervised link embedding framework, which utilizes the strong correlation between central links and their neighborhood subgraphs to characterize links. We extract the "link-centric induced subgraphs" as input, with a subgraph-level contrastive discrimination as pretext task, to learn the intrinsic and structural link features via subgraph mini-batch training. Extensive experiments conducted on five datasets demonstrate that SCLRL has significant performance advantages in link representation learning on benchmark datasets and prominent efficiency advantages in terms of training speed and memory consumption on large-scale graphs, when compared with existing link representation learning methods.