Assuming you're taking a scientific stance and not just blurting out what all insulin and lipolysis low carbers parrot; every time you eat there is a change in adipose tissue TG lipolysis via the actions of insulin. cAMP-dependent changes that occur in response to insulin binding are effected by activation of phosphodiesterase which hydrolyzes cAMP rendering PKA much less active. activation of phosphodiesterase 3B occurs via PKB/Akt-mediated phosphorylation which itself is activated following the insulin binding of its receptor.
The mechanism for insulin mediated reduction in TG lipolysis is due to the stimulation of phosphatase-1 which removes the phosphate from hormone sensitive lipase making it much less active. The activity of hormone sensitive lipase would also affected by phosphorylation of AMPK, so, the phosphorylation would inhibit. Inhibition of hormone sensitive lipase by AMPK might seem like it would be counterproductive because the release of fatty acids stored in TG would seem necessary to promote the production of ATP via fatty acid oxidation and the major function of AMPK is to shift cells to ATP production from ATP consumption(this is seen in the increased NAD+ production in fatty acid oxidation). This is explained when you realize that if the fatty acids that are released from TG are not consumed they will be recycled back into TGs at the expense of ATP consumption, so, inhibition of hormone sensitive lipase by AMPK mediated-phosphorylation is a mechanism to ensure that the rate of fatty acid release does not exceed the rate at which they are utilized either by exporting it out or oxidation(fat loss).
Lipolysis creates excess NEFAs by design, it is not the sole regulation of weight loss. It's also a short blunt anyway as all things are regulated by rate-limiting steps.