what is the influence to R square when more independent

having more covariates will in general give a better fit, however, this does not necessarily mean a better model (e.g., in terms of generalization). Thus, model comparison is carried out at the end in terms of (1) how the model explains the data (e.g., likelihood, R2, etc), and (2) how simple the model is (i.e., Occam's razor)

One could get a better R squared for having more variables, but that potentially induces overfitting for the sample of data you are looking at. One can look at BIC where the model selection criterion is penalized by having too many variables.