Humans can effortlessly modify various prosodic attributes, such as the placement of stress and the intensity of sentiment, to convey a specific emotion while maintaining consistent linguistic content. Motivated by this capability, we propose EmoAug, a novel style transfer model designed to enhance emotional expression and tackle the data scarcity issue in speech emotion recognition tasks. EmoAug consists of a semantic encoder and a paralinguistic encoder that represent verbal and non-verbal information respectively. Additionally, a decoder reconstructs speech signals by conditioning on the aforementioned two information flows in an unsupervised fashion. Once training is completed, EmoAug enriches expressions of emotional speech with different prosodic attributes, such as stress, rhythm and intensity, by feeding different styles into the paralinguistic encoder. EmoAug enables us to generate similar numbers of samples for each class to tackle the data imbalance issue as well. Experimental results on the IEMOCAP dataset demonstrate that EmoAug can successfully transfer different speaking styles while retaining the speaker identity and semantic content. Furthermore, we train a SER model with data augmented by EmoAug and show that the augmented model not only surpasses the state-of-the-art supervised and self-supervised methods but also overcomes overfitting problems caused by data imbalance. Some audio samples can be found on our demo website.