Please check these useful links

http://www.sciencedirect.com/science/article/pii/S0944501313001432

http://www.sciencedirect.com/science/article/pii/S2210784316301346

http://link.springer.com/chapter/10.1007/978-81-322-2776-2_21

http://link.springer.com/article/10.1007/s11104-008-9805-z#page-1

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mainly by weathering minerals through acidic exudates.

Potassium solubilisation takes place through  organic acids ( mainly oxalic acid , ferulic acid etc )  released by bacteria as well as plant roots as exudates acted upon insoluble K-bearing minerals.

potassium solubilizing bacteria convert insoluble potassium to soluble form for plants :

first : by acidification trhough the production of organique acide , its the same thing pronunced by Dr.Anoop Kumar Srivastava.

second : by chelation of silicate and aluminium and releasing soluble forme of potassium. 

The mechanisms for bacteria to solubilization of K are by many ways:

1. lowering the pH, or

2. by enhancing chelation of the cations bound to K, and

3. acidolysis of the surrounding area of microorganism.

The lowering in pH of the medium suggests the release of organic acids and protons by the K-solubilizing microorganisms.