From the Super Mario Wiki
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The main purpose of a Nasty Net is for catching and holding enemy targets, such as Luigi. Upon hitting its target, a Nasty Net will wrap itself around it entirely.
They are fairly durable, and do not conduct electricity, as Mario and Luigi managed to collect a school of Jelectros in one without getting shocked. While most characters are not strong enough to destroy a Nasty Net, a Chain Chomp can chew through them in a single bite.