Unmanned aerial vehicles (UAV) has been increasingly applied to crop growth monitoring due to their advantages, such as their rapid and repetitive capture ability, high resolution, and low cost. LAI is an important parameter for evaluating crop canopy structure and growth without damage. Accurate monitoring of cotton LAI has guiding significance for nutritional diagnosis and the accurate fertilization of cotton. This study aimed to obtain hyperspectral images of the cotton canopy using a UAV carrying a hyperspectral sensor and to extract effective information to achieve cotton LAI monitoring. In this study, cotton field experiments with different nitrogen application levels and canopy spectral images of cotton at different growth stages were obtained using a UAV carrying hyperspectral sensors. Hyperspectral reflectance can directly reflect the characteristics of vegetation, and vegetation indices (VIs) can quantitatively describe the growth status of plants through the difference between vegetation in different band ranges and soil backgrounds. In this study, canopy spectral reflectance was extracted in order to reduce noise interference, separate overlapping samples, and highlight spectral features to perform spectral transformation; characteristic band screening was carried out; and VIs were constructed using a correlation coefficient matrix. Combined with canopy spectral reflectance and VIs, multiple stepwise regression (MSR) and extreme learning machine (ELM) were used to construct an LAI monitoring model of cotton during the whole growth period. The results show that, after spectral noise reduction, the bands screened by the successive projections algorithm (SPA) are too concentrated, while the sensitive bands screened by the shuffled frog leaping algorithm (SFLA) are evenly distributed. Secondly, the calculation of VIs after spectral noise reduction can improve the correlation between vegetation indices and LAI. The DVI (540,525) correlation was the largest after standard normal variable transformation (SNV) pretreatment, with a correlation coefficient of −0.7591. Thirdly, cotton LAI monitoring can be realized only based on spectral reflectance or VIs, and the ELM model constructed by calculating vegetation indices after SNV transformation had the best effect, with verification set R2 = 0.7408, RMSE = 1.5231, and rRMSE = 24.33%, Lastly, the ELM model based on SNV-SFLA-SNV-VIs had the best performance, with validation set R2 = 0.9066, RMSE = 0.9590, and rRMSE = 15.72%. The study results show that the UAV equipped with a hyperspectral sensor has broad prospects in the detection of crop growth index, and it can provide a theoretical basis for precise cotton field management and variable fertilization.