Recently, recommender systems have received an increasing amount of attention from researchers due to their indispensable role in the more and more popular e-commercial websites. Although a lot of methods have been proposed for warm-start recommendation, cold-start recommendation still remains open as one of the major challenges of recommender systems. The existing approaches often suffer from two defects. The first is the lack of unified framework. The existing researches often deal with the cold-start recommendation and the warm-start recommendation separately, which makes their respective methods hard to integrate into a system and keeps the cold-start users/items away from the existing ones. The second is the poor interpretability. The existing methods often ignore the complicated preferential relationships between users and item features, and can not quantitatively explain the multiple reasons that cause a user chooses an item. In this paper, we aim at the problem of making explainable recommendations for both warm-start and cold-start users/items in a unified framework, of which the challenges are three-fold, the lack of meaningful information, large-scale data, and quantitative explanation. To address these challenges, we propose a novel concept referred to as meta-feature, and a Meta-feature based Explainable Recommendation Framework (MERF). Meta-features are latent features about item features, which can reveal the preferential relationship between users and item features, not just the items. MERF is able to make recommendations for both cold-start and warm-start users/items in a unified framework based on meta-feature. Especially, thanks to meta-feature, MERF can make cold-start recommendations requiring no historical rating records but just the item features. To make a recommendation with a quantitative explanation, we propose a Personalized Feature Preference (PFP) vector to characterize the different importance of item features to a user. MERF makes a recommendation based on an Item Rating Matrix and an Explanation Matrix, which can be estimated by fusing PFP and meta-features. To improve the efficiency of MERF, we also propose a parallel learning algorithm and an incremental updating algorithm for PFP. At last, extensive experiments conducted on real datasets verify the effectiveness and efficiency of the proposed approach.