Can someone help me my confusion about the Bayesian estimation of DSGE model below?

In Dynare we specify

Blockquote mode_compute

is to find the mode computation. I wonder that this “mode_compute” is to find the maximum likelihood or what? Because the posterior density is proportional to the product of prior and likelihood densities, thus, the finding of the mode means that finding of the highest degree of the log-likelihood.
Once the mode is found, the next step is to use the MCMC for inference

No, unless the prior is uniform over the parameter space, maximizing the posterior density is not equivalent to maximizing the likelihood.

The mode is one summary statistic of the distribution. We use the MCMC to get other statistics of interest like mean, standard deviation, percentiles, etc.

could you kindly explane it more detaile for what is the maximization of the posterior, how does it work?
My understanding in the beginning is that since posterior is propotional to the product of the prior and the likelihood, but the prior is fixed, that’s why i thinkg the maximization of the posterior is equavalent to the maximization of the likelihood?

You are trying to find the parameter vector that maximizes the posterior. The prior is a density over the parameter space. Say a parameter rho has a standard normal distribution as its prior. If you vary the value of rho, obviously the normal density at this value varies. Hence, the prior density is not a constant, unless you use a uniform distribution.