Hormone sensitive lipase (HSL) is the enzyme which degrades triglycerides stored within adipocytes (FA2020 p93). Thus, it makes sense that it is activated in times of fasting and suppressed in the fed state.
Insulin would inhibit HSL, as insulin is a fed state enzyme secreted by the pancreas and would want to trigger storage of triglycerides.
In contrast glucagon is secreted in response to hypoglycemia by the pancreas and will trigger fasted state activation. In terms of the fed/fast state I always think of glucagon and epinephrine kind of like a superhero and their side kick, because they usually work together in the fasting state on similar targets to ensure the body has enough energy (this helps me remember that epinephrine and glucagon are fasting state hormones). Here though is epinephrine's big action away from glucagon, where glucagon has minimal effect and epinephrine has the big action of activating HSL! Glucagon has a minor role and other catecholamines and ACTH can also serve to activate HSL as well.
Another example of the synergistic work of glucagon and epinephrine is in glycogen breakdown (FA2020 p85). Both will trigger cAMP increase and protein kinase A activation which will phosphorylate glycogen phosphporylase and activate it (FAST PHOSPHORYLATE! Hormone sensitive lipase is actually phsophorylated to activate it as well).
FUN FACT: Hormone sensitive lipase actually got its name because it was sensitive to epinephrine!