Forecasting the generation of solar power plants (SPPs) requires taking into account meteorological parameters that influence the difference between the solar irradiance at the top of the atmosphere calculated with high accuracy and the solar irradiance at the tilted plane of the solar panel on the Earth’s surface. One of the key factors is cloudiness, which can be presented not only as a percentage of the sky area covered by clouds but also many additional parameters, such as the type of clouds, the distribution of clouds across atmospheric layers, and their height. The use of machine learning algorithms to forecast the generation of solar power plants requires retrospective data over a long period and formalising the features; however, retrospective data with detailed information about cloudiness are normally recorded in the natural language format. This paper proposes an algorithm for processing such records to convert them into a binary feature vector. Experiments conducted on data from a real solar power plant showed that this algorithm increases the accuracy of short-term solar irradiance forecasts by 5–15%, depending on the quality metric used. At the same time, adding features makes the model less transparent to the user, which is a significant drawback from the point of view of explainable artificial intelligence. Therefore, the paper uses an additive explanation algorithm based on the Shapley vector to interpret the model’s output. It is shown that this approach allows the machine learning model to explain why it generates a particular forecast, which will provide a greater level of trust in intelligent information systems in the power industry.