Usually this means one of two things:

1. The colonies you selected were DH5a that spontaneously acquired mutations that gave them a low level of kanamycin resistance, but were not actually transformed with the plasmid. Sometimes these bacteria end up with odd growth phenotypes in broth. Often you can solve this problem by remaking old kanamycin plates and/or increasing the concentration of the kanamycin in the agar.

2. Some random environmental microbes are contaminating your sample and what's growing isn't even E. coli. There's multiple places where these organisms can get in (ie. contaminated broth), but a common point of contamination, when proper sterile technique has otherwise been applied, is the actual plasmid itself. It's rare that DNA elution buffers and enzymatic reactions are kept completely sterile. Usually this doesn't matter because any contaminants are killed by the selection antibiotic. But not always.

In addition to the good answers provide by Alexandra Johnson you may be seeing lysis. The fact that you are seeing "thread like structures" is highly suggestive that the culture is lysing. So another possibility could be that you have a phage contamination.