The starting point is to identify classical liberalism as a veneer over power structures based on arbitrary authority. For example, the same society that claimed to support freedom of expression would lock women away in mental institutions for acting in a way deemed inappropriate (not so much nowadays, but definitely in the 30's). Today we still see a lot of arbitrary authority (often illegal) exercised by the government and its agents. Only a small portion of this is deemed "political" enough to gain media attention, and perhaps be corrected in the courts.
Furthermore, the ideology of individual rights can be used to create "rights" that couldn't even be justified from a natural rights perspective, but are rather created to bolster the idea of an atomized society. Government subsidization of roads (rather than public transport) is a good example of this. While there is no natural right to drive a car on a government subsidized road, the ideology of individualism leads people to suppose that people are somehow more "free" under this setup than under a regime of mainly public transportation.
To summarize, left-anarchists view classical individual rights as a chimera, where the real agenda is a society of atomized individuals with no sense of community, all subject to the authority of the state. The main claim of left-anarchists is that in the absence of this oppression, a natural order based on community enforcement will evolve, in which strict legal rights are not necessary to protect individual's wellbeing, because the community is good and cares about its members. Various examples are given for this, such as primitive tribal societies, temporary anarchistic situations such as during the Spanish civil war, working class communities who self-police rather than cooperating with law enforcement, and social orders that arise around the use of common property, such as with surfing.