Restrained by the long term evolution (LTE) limited network capacity, WiFi technology is considered as one of the promising solutions to leverage the traffic load and enhance the LTE capacity. Exploiting both the licensed and unlicensed spectrum was the motivated key to standardize the LTE-wireless local area network (WLAN) aggregation (LWA) technology by 3GPP in Release 13. In this paper, we consider the user association problem in LWA-based Multiple Radio Access Technologies (Multi-RAT) Heterogeneous Networks (HetNet) in which three transmission modes are available (LTE, WiFi, and aggregation mode) and the user needs to select not only the wireless node that will associate with it, but also the used transmission mode. For this, a new user association algorithm that considers the joint node and mode selection is proposed in this paper. This association process is formulated as an optimization problem with the aim to maximize total network throughput. To solve this problem, a one-to-many matching game-based association algorithm is designed, where each user is matched to the best transmission mode/node according to well-developed utility function that considers the achieved data rate of each user as well as the proportional fairness among users. Simulation results have shown that our proposed algorithm outperforms comparable association techniques such as WLAN first, LTE first, and LTE-W in terms of system throughput, outage probability and fairness between users.