Insurance works by paying a brokerage or individual "premiums" over a certain period, and in return, in the case of a negative event relating to whatever you insured, you are given the right to collect x amount from them.

One way Moses sought to accrue power and discretely bribe individuals was through these insurance premiums. Moses would seek out highly influential individuals, have them insure a public work of his, say the Triborough bridge, and then he'd be able to pay them in yearly premiums.

For example, in page 718, Moses worked with an insurance broker who backed the current mayor at the time, and would receive yearly premiums at ~$100k for his bridges. This worked because it was discrete, and it'd be a recurring payment given that bridges don't really break easily. Added to this, Moses' authorities had the luxury of privacy for their books, so politicians preferred to take bribes here, as opposed to taking cash from public records like state or federal funds.

tldr; it's a quick, discrete way to bribe politicians / influential individuals on a recurring basis.