Normally, leptin's function is to reduce appetite and induce fat burning (among many other functions). That is what high leptin signaling in a brain would do. Low leptin (in the brain) is an indication to eat more and store more fat (to successfully reproduce and to live long enough to do so).
However, elevated leptin in a fasting blood sample indicates leptin resistance and likely low leptin signaling to some parts of the brain while other parts of the brain get the full high signal. In other words, some of the brain only hears a whisper while other parts (of the brain and periphery) get screamed at.
Neither is Good Communication
Low leptin signaling getting through to the appetite center of the brain induces the brain to want to make the rest of your body hungry and will alter physiologic functions so as to make you store more fat. Ultimately, and finally, increasing fat stores should manufacture more leptin to overcome the resistance but, in the meantime, one continues to get fat and often ultimately obese.
This is similar to insulin resistance, when high fasting insulin indicates low activity in some parts of your body and a disruption in insulin signaling that is being compensated for by your pancreas making more insulin. What is lost, however, is your orchestration of insulin levels among various tissues. If your liver is insulin- resistant, it continues to make sugar out of protein, and if your muscles are insulin-resistant, they cannot burn that sugar either.
However, until your fat tissue becomes insulin-resistant, it continues to "hear" the high levels of insulin that are caused by the elevated sugar, and insulin's signals to fat tissue is to take that sugar, make fat out of it, and then store it. The positive side of this is that you're still able to take sugar out of the blood to make fat out of it. This keeps you from becoming diabetic, at least in the short-term.
In this regard, one could say that obesity is the price one pays to keep from becoming diabetic. One continues to gain weight until the adipose tissue ultimately becomes resistant. At this time, your "wastebasket" to store the excess sugar becomes full, sugar accumulates in your blood, and conventional medicine will diagnose you as a diabetic, though the root problem of insulin resistance and perhaps more importantly, leptin resistance, began decades prior (perhaps even before you were born if your mother was feeding you lots of sugar/starches when you were a fetus).