This may not be the only method (there may be another more specific mechanism) but tannins sequester some metal ions such as iron, lowering their availability. I understand that this can act as a bacteriostatic against organisms without a strong siderophore system. The papers in question were published in the early 1990s, if I recall correctly.

Also, tannins can crosslink proteins, and thereby act as inhibitors to protein digestion and function. See http://www.ncbi.nlm.nih.gov/pubmed/7665384 to get you started on this.

Most of the tannins are feeding deterrent to insects but their formulations if provided in artificial diet may provide stomach poisoning, normally tannins show systemic effects in insects and absorb through epidermis. 

Tannins are polymers of phenolic compounds and are deeding deterrents which reduce the digestibility of food in insect gut. They can also bind to certain quinones and form complexes which ultimately cause mortality in insects over a period of time.

Please see the following article: Scalbert A. (1991). Antimicrobial properties of tannins. Phytochemistry, 30, 3875-3883.

I hope this can help you.

http://www.sciencedirect.com/science/article/pii/003194229183426L

 Dear What is column we can use to separate tannins from crude extracts 

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