Despite the evidence of corruption, political machines retained strong public support. Why did people continue to support the machines?
The two main reasons why people continued to support political machines was because the machines A) provided them with material benefits in return for support and B) because the machines could punish many of them for failing to support the machines.
(Assuming we're talking about the early 1900s here...)
Many of the machines' constituents were relatively poor immigrants. The machines provided them with jobs and with help when they needed it. Today, most people don't depend on elected officials to directly give them jobs or benefits, but back then, many people did (because almost all city jobs back then were controlled by the mayors, who could hire and fire people even without a good reason).
Furthermore,these kinds of help could be withheld if the people receiving them failed to vote for the machines (remember, there was no secret ballot back in those days). For example, if you had a city job, you might be expected to make sure that 90% of the people in your precinct voted the right way. If they didn't, you lost your job and they (the people) didn't get benefits anymore.
So, people continued to support machines because they got tangible benefits for supporting the machines and they could lose those benefits if they didn't support the machines.
In the early 1900's society was so much simplier than it is now. The local politicians were much more intimate with their constituents. They provided the poor immigrants with jobs. They helped the more established business owners get political favours, to ensure their business success. Police protected businesses where they received money under the table or where the political bosses were indebited to the business owners. Graft was the usual way. No one would go aginst them because they were afraid; the consequences were too great to take a chance.
The politicians were much more intimate, or had much influence, in a sense, over their lives. They could help to obtain political favor, get jobs or ensured that the citizens were protected. This was too much to risk for the removal of the machines, therefore, the politcal machines remained.