Why people respond to surveys like that of Purchasing Managers' Index? A lot of data, including indexes, rely on surveys. For example, Purchasing Managers' Index.

(1) Why people answer these surveys (is there any benefit)?

(2) How to make sure the data is correct? What if people randomly fill in the survey, or intentionally give incorrect data?
For your first question:

Some have noted that surveys make people feel like they are a part of something bigger than themselves and that their opinion matters. 

According to Ed Halteman (survey expert) two reasons why people participate in surveys, ,  are that they feel their participation will affect something they care about, and they want to share their opinion with someone who will listen and act on the information.

For your second question:

Once the surveys(or census) are collected there are two stages that typically takes place to ensure that the data products are of high quality.  They are data processing and statistical adjustments. 

In data processing, a survey statistician or analysts,   use codes to capture data from their raw state to an edited data file that can be used within the survey organization for quality assessment in preparation for statistical adjustment and dissemination.

They use coding to classify survey responses into categories with associated numeric values. They also decide how coding and data capture will be conducted and monitored. A survey statistician or analysts will edit the data to check for errors throughout the survey lifecycle. 

Statistical adjustments:  Consideration will be made for missing data or data values that are obviously wrong using single or multiple imputation methods.  

Analysts will use codes to calculate outcome rates based on available data according to the agencies methodology(s).  They will also develop survey weights for each interviewed element on the sampling frame.  

Throughout the above processes, documentation is done during both data processing and statistical adjustment steps.