LH (luteinizing hormone) and FSH (follicle-stimulating hormone) are gonadotropins: hormones released by the pituitary gland, under direction from the hypothalamus, that control the testes and ovaries. In men, LH tells the testes to make testosterone and FSH supports sperm production. In women, FSH grows the ovarian follicles that hold eggs, and a mid-cycle surge of LH triggers ovulation. Doctors measure them in blood tests to work out where a hormone problem starts: low testosterone with low or normal LH points to the brain-pituitary signal, while high LH points to the testes themselves.
They matter on a GLP-1 for a few reasons. Obesity can suppress the LH signal in men, which is one reason testosterone often rises with weight loss. Taking testosterone therapy does the opposite, shutting down LH and FSH, which is why it can reduce sperm production. In women, a high LH-to-FSH ratio is common in PCOS, and ovulation can return as weight comes off. See men's health on a GLP-1 and women's health on a GLP-1.
Source: MedlinePlus (external link).