in the following table (association between cognitive score and Folate intake), the beta coefficient represents difference in slope between different groups with the standard group. the interpretation of such data will present the change in the dependent value per unit change of the independent value? or something else? and the second question, is it appropriate here to use the linear regression not the logistic regression? or both of them are feasible?
Whether they should use linear regression or logistic regression depends mostly on whether the dependent variable (which you say is cognitive score) is continuous or not. If it is continuous linear regression would be the usual starting place. If it is discrete then the usual starting place would be logistic or maybe ordinal logistic.
As to what the betas are, it looks like they divided folate intake by quintile, then used it as a categorical variable in the regression and then did some test for trend. So each beta compares a particular quintile of folate intake to the lowest quintile.
This is a not uncommon method, but I don't really approve of it. I'd suggest that a better analysis would be to leave folate intake continuous and then use a spline or something similar. But I can't say it's absolutely wrong to do what they did.